https://hilbert.math.wisc.edu/wiki/api.php?action=feedcontributions&user=Dummit&feedformat=atomUW-Math Wiki - User contributions [en]2021-10-20T16:22:08ZUser contributionsMediaWiki 1.30.1https://hilbert.math.wisc.edu/wiki/index.php?title=NTSGrad_Spring_2014&diff=6456NTSGrad Spring 20142014-01-23T20:08:28Z<p>Dummit: /* Spring 2014 Semester */</p>
<hr />
<div>= Number Theory – Representation Theory Graduate Student Seminar, University of Wisconsin–Madison =<br />
<br />
*'''When:''' Tuesdays, 2:30pm–3:30pm<br />
*'''Where:''' Van Vleck B131<br />
<br />
The purpose of this seminar is to have a talk on each Tuesday by a graduate student to<br />
help orient ourselves for the [[NTS|Number Theory Seminar]] talk on the following Thursday.<br />
These talks should be aimed at beginning graduate students, and should try to <br />
explain some of the background, terminology, and ideas for the Thursday talk. <br />
<br />
== Spring 2014 Semester ==<br />
<br />
<center><br />
<br />
{| style="color:black; font-size:120%" border="0" cellpadding="14" cellspacing="5"<br />
|-<br />
| bgcolor="#D0D0D0" width="300" align="center"|'''Date'''<br />
| bgcolor="#F0A0A0" width="300" align="center"|'''Speaker'''<br />
| bgcolor="#BCD2EE" width="300" align="center"|'''Title (click to see abstract)'''<br />
|- <br />
| bgcolor="#E0E0E0"| Jan 21 (Tues.)<br />
| bgcolor="#F0B0B0"| No talk<br />
| bgcolor="#BCE2FE"|[[NTS Spring 2014/Abstracts | <font color="black"><em>--</em></font>]]<br />
|- <br />
| bgcolor="#E0E0E0"| Jan 28 (Tues.)<br />
| bgcolor="#F0B0B0"| who?<br />
| bgcolor="#BCE2FE"|[[NTS Spring 2014/Abstracts | <font color="black"><em>tba</em></font>]]<br />
|- <br />
| bgcolor="#E0E0E0"| Feb 4 (Tues.)<br />
| bgcolor="#F0B0B0"| who?<br />
| bgcolor="#BCE2FE"|[[NTS Spring 2014/Abstracts | <font color="black"><em>tba</em></font>]]<br />
|- <br />
| bgcolor="#E0E0E0"| Feb 11 (Tues.)<br />
| bgcolor="#F0B0B0"| who?<br />
| bgcolor="#BCE2FE"|[[NTS Spring 2014/Abstracts | <font color="black"><em>tba</em></font>]]<br />
|- <br />
| bgcolor="#E0E0E0"| Feb 18 (Tues.)<br />
| bgcolor="#F0B0B0"| who?<br />
| bgcolor="#BCE2FE"|[[NTS Spring 2014/Abstracts | <font color="black"><em>tba</em></font>]]<br />
|- <br />
| bgcolor="#E0E0E0"| Feb 25 (Tues.)<br />
| bgcolor="#F0B0B0"| who?<br />
| bgcolor="#BCE2FE"|[[NTS Spring 2014/Abstracts | <font color="black"><em>tba</em></font>]]<br />
|- <br />
| bgcolor="#E0E0E0"| Mar 4 (Tues.)<br />
| bgcolor="#F0B0B0"| who?<br />
| bgcolor="#BCE2FE"|[[NTS Spring 2014/Abstracts | <font color="black"><em>tba</em></font>]]<br />
|- <br />
| bgcolor="#E0E0E0"| Mar 11 (Tues.)<br />
| bgcolor="#F0B0B0"| who?<br />
| bgcolor="#BCE2FE"|[[NTS Spring 2014/Abstracts | <font color="black"><em>tba</em></font>]]<br />
|- <br />
| bgcolor="#E0E0E0"| Mar 18 (Tues.)<br />
| bgcolor="#F0B0B0"| -----<br />
| bgcolor="#BCE2FE"|[[NTS Spring 2014/Abstracts | <font color="black"><em>spring break</em></font>]]<br />
|- <br />
| bgcolor="#E0E0E0"| Mar 25 (Tues.)<br />
| bgcolor="#F0B0B0"| who?<br />
| bgcolor="#BCE2FE"|[[NTS Spring 2014/Abstracts | <font color="black"><em>tba</em></font>]]<br />
|-<br />
| bgcolor="#E0E0E0"| Apr 1 (Tues.)<br />
| bgcolor="#F0B0B0"| who?<br />
| bgcolor="#BCE2FE"|[[NTS Spring 2014/Abstracts | <font color="black"><em>tba</em></font>]]<br />
|-<br />
| bgcolor="#E0E0E0"| Apr 8 (Tues.)<br />
| bgcolor="#F0B0B0"| who?<br />
| bgcolor="#BCE2FE"|[[NTS Spring 2014/Abstracts | <font color="black"><em>tba</em></font>]]<br />
|- <br />
| bgcolor="#E0E0E0"| Apr 15 (Tues.)<br />
| bgcolor="#F0B0B0"| who?<br />
| bgcolor="#BCE2FE"|[[NTS Spring 2014/Abstracts | <font color="black"><em>tba</em></font>]]<br />
|- <br />
| bgcolor="#E0E0E0"| Apr 22 (Tues.)<br />
| bgcolor="#F0B0B0"| who?<br />
| bgcolor="#BCE2FE"|[[NTS Spring 2014/Abstracts | <font color="black"><em>tba</em></font>]]<br />
|- <br />
| bgcolor="#E0E0E0"| Apr 29 (Tues.)<br />
| bgcolor="#F0B0B0"| who?<br />
| bgcolor="#BCE2FE"|[[NTS Spring 2014/Abstracts | <font color="black"><em>tba</em></font>]]<br />
|- <br />
| bgcolor="#E0E0E0"| May 6 (Tues.)<br />
| bgcolor="#F0B0B0"| who?<br />
| bgcolor="#BCE2FE"|[[NTS Spring 2014/Abstracts | <font color="black"><em>tba</em></font>]]<br />
|}<br />
<br />
</center><br />
<br />
<br><br />
<br />
== Organizers ==<br />
<br />
[http://www.math.wisc.edu/~rharron/ Robert Harron]<br />
<br />
Sean Rostami<br />
<br />
----<br />
The Fall 2013 NTS Grad page can be found [[NTSGrad Fall 2013|here]].<br />
----<br />
Return to the [[NTS|Number Theory Seminar Page]]<br />
<br />
Return to the [[Algebra|Algebra Group Page]]</div>Dummithttps://hilbert.math.wisc.edu/wiki/index.php?title=NTSGrad_Spring_2014&diff=6455NTSGrad Spring 20142014-01-23T20:06:36Z<p>Dummit: /* Spring 2014 Semester */</p>
<hr />
<div>= Number Theory – Representation Theory Graduate Student Seminar, University of Wisconsin–Madison =<br />
<br />
*'''When:''' Tuesdays, 2:30pm–3:30pm<br />
*'''Where:''' Van Vleck B131<br />
<br />
The purpose of this seminar is to have a talk on each Tuesday by a graduate student to<br />
help orient ourselves for the [[NTS|Number Theory Seminar]] talk on the following Thursday.<br />
These talks should be aimed at beginning graduate students, and should try to <br />
explain some of the background, terminology, and ideas for the Thursday talk. <br />
<br />
== Spring 2014 Semester ==<br />
<br />
<center><br />
<br />
{| style="color:black; font-size:120%" border="0" cellpadding="14" cellspacing="5"<br />
|-<br />
| bgcolor="#D0D0D0" width="300" align="center"|'''Date'''<br />
| bgcolor="#F0A0A0" width="300" align="center"|'''Speaker'''<br />
| bgcolor="#BCD2EE" width="300" align="center"|'''Title (click to see abstract)'''<br />
|- <br />
| bgcolor="#E0E0E0"| Jan 21 (Tues.)<br />
| bgcolor="#F0B0B0"| No talk<br />
| bgcolor="#BCE2FE"|[[NTS Spring 2014/Abstracts | <font color="black"><em>--</em></font>]]<br />
|- <br />
| bgcolor="#E0E0E0"| Jan 28 (Tues.)<br />
| bgcolor="#F0B0B0"| who?<br />
| bgcolor="#BCE2FE"|[[NTS Spring 2014/Abstracts | <font color="black"><em>tba</em></font>]]<br />
|- <br />
| bgcolor="#E0E0E0"| Feb 4 (Tues.)<br />
| bgcolor="#F0B0B0"| who?<br />
| bgcolor="#BCE2FE"|[[NTS Spring 2014/Abstracts | <font color="black"><em>tba</em></font>]]<br />
|- <br />
| bgcolor="#E0E0E0"| Feb 11 (Tues.)<br />
| bgcolor="#F0B0B0"| who?<br />
| bgcolor="#BCE2FE"|[[NTS Spring 2014/Abstracts | <font color="black"><em>tba</em></font>]]<br />
|- <br />
| bgcolor="#E0E0E0"| Feb 18 (Tues.)<br />
| bgcolor="#F0B0B0"| who?<br />
| bgcolor="#BCE2FE"|[[NTS Spring 2014/Abstracts | <font color="black"><em>tba</em></font>]]<br />
|- <br />
| bgcolor="#E0E0E0"| Feb 25 (Tues.)<br />
| bgcolor="#F0B0B0"| who?<br />
| bgcolor="#BCE2FE"|[[NTS Spring 2014/Abstracts | <font color="black"><em>tba</em></font>]]<br />
|- <br />
| bgcolor="#E0E0E0"| Mar 4 (Tues.)<br />
| bgcolor="#F0B0B0"| who?<br />
| bgcolor="#BCE2FE"|[[NTS Spring 2014/Abstracts | <font color="black"><em>tba</em></font>]]<br />
|- <br />
| bgcolor="#E0E0E0"| Mar 11 (Tues.)<br />
| bgcolor="#F0B0B0"| who?<br />
| bgcolor="#BCE2FE"|[[NTS Spring 2014/Abstracts | <font color="black"><em>tba</em></font>]]<br />
|- <br />
| bgcolor="#E0E0E0"| Mar 18 (Tues.)<br />
| bgcolor="#F0B0B0"| spring break!<br />
| bgcolor="#BCE2FE"|[[NTS Spring 2014/Abstracts | <font color="black"><em>spring break</em></font>]]<br />
|- <br />
| bgcolor="#E0E0E0"| Mar 25 (Tues.)<br />
| bgcolor="#F0B0B0"| who?<br />
| bgcolor="#BCE2FE"|[[NTS Spring 2014/Abstracts | <font color="black"><em>tba</em></font>]]<br />
|-<br />
| bgcolor="#E0E0E0"| Apr 1 (Tues.)<br />
| bgcolor="#F0B0B0"| who?<br />
| bgcolor="#BCE2FE"|[[NTS Spring 2014/Abstracts | <font color="black"><em>tba</em></font>]]<br />
|-<br />
| bgcolor="#E0E0E0"| Apr 8 (Tues.)<br />
| bgcolor="#F0B0B0"| who?<br />
| bgcolor="#BCE2FE"|[[NTS Spring 2014/Abstracts | <font color="black"><em>tba</em></font>]]<br />
|- <br />
| bgcolor="#E0E0E0"| Apr 15 (Tues.)<br />
| bgcolor="#F0B0B0"| who?<br />
| bgcolor="#BCE2FE"|[[NTS Spring 2014/Abstracts | <font color="black"><em>tba</em></font>]]<br />
|- <br />
| bgcolor="#E0E0E0"| Apr 22 (Tues.)<br />
| bgcolor="#F0B0B0"| who?<br />
| bgcolor="#BCE2FE"|[[NTS Spring 2014/Abstracts | <font color="black"><em>tba</em></font>]]<br />
|- <br />
| bgcolor="#E0E0E0"| Apr 29 (Tues.)<br />
| bgcolor="#F0B0B0"| who?<br />
| bgcolor="#BCE2FE"|[[NTS Spring 2014/Abstracts | <font color="black"><em>tba</em></font>]]<br />
|- <br />
| bgcolor="#E0E0E0"| May 6 (Tues.)<br />
| bgcolor="#F0B0B0"| who?<br />
| bgcolor="#BCE2FE"|[[NTS Spring 2014/Abstracts | <font color="black"><em>tba</em></font>]]<br />
|}<br />
<br />
</center><br />
<br />
<br><br />
<br />
== Organizers ==<br />
<br />
[http://www.math.wisc.edu/~rharron/ Robert Harron]<br />
<br />
Sean Rostami<br />
<br />
----<br />
The Fall 2013 NTS Grad page can be found [[NTSGrad Fall 2013|here]].<br />
----<br />
Return to the [[NTS|Number Theory Seminar Page]]<br />
<br />
Return to the [[Algebra|Algebra Group Page]]</div>Dummithttps://hilbert.math.wisc.edu/wiki/index.php?title=741&diff=62847412013-11-18T23:39:08Z<p>Dummit: /* HOMEWORK 9 (Due Nov 13) */</p>
<hr />
<div>'''Math 741'''<br />
<br />
Algebra<br />
<br />
Prof: [http://www.math.wisc.edu/~ellenber Jordan Ellenberg]<br />
<br />
Grader: Evan Dummit.<br />
* Homework policies:<br />
* Late homework may be given directly to the grader, along with either <br />
(i) the instructor's permission, or (ii) a polite request for mercy.<br />
* Assignments that are more than 1 page should be affixed in some reasonable way.<br />
* Results from places (e.g., the internet) other than 741 and standard books must be cited.<br />
<br />
JE's office hours: Monday 12pm-1pm (right after class)<br />
<br />
This course, the first semester of the introductory graduate sequence in algebra, will cover the basic theory of groups, group actions, representation, linear and multilinear algebra, and the beginnings of ring theory.<br />
<br />
==SYLLABUS==<br />
<br />
In this space we will record the theorems and definitions we covered each week, which we can use as a list of notions you should be prepared to answer questions about on the Algebra qualifying exam. The material covered on the homework is also an excellent guide to the scope of the course. <br />
<br />
'''WEEK 1''': <br />
<br />
Definition of group. Associativity. Inverse. <br />
<br />
Examples of group: GL_n(R). GL_n(Z). Z/nZ. R. Z. R^*. The free group F_k on k generators. <br />
<br />
Homomorphisms. The homomorphisms from F_k to G are in bijection with G^k. Isomorphisms.<br />
<br />
'''WEEK 2''':<br />
<br />
The symmetric group (or permutation group) S_n on n letters. Cycle decomposition of a permutation. Order of a permutation. Thm: every element of a finite group has finite order. <br />
<br />
Subgroups. Left and right cosets. Lagrange's Theorem. Cyclic groups. The order of an element of a finite group is a divisor of the order of the group.<br />
<br />
The sign homomorphism S_n -> +-1.<br />
<br />
<br />
'''WEEK 3'''<br />
<br />
Normal subgroups. The quotient of a group by a normal subgroup. The first isomorphism theorem. Examples of S_n -> +-1 and S_4 -> S_3 with kernel V_4, the Klein 4-group.<br />
<br />
Centralizers and centers. Abelian groups. The center of SL_n(R) is either 1 or +-1.<br />
<br />
Groups with presentations. The infinite dihedral group <x,y | x^2 = 1, y^2 = 1>.<br />
<br />
'''WEEK 4'''<br />
<br />
More on groups with presentations.<br />
<br />
Second and third isomorphism theorems.<br />
<br />
Semidirect products.<br />
<br />
'''WEEK 5'''<br />
<br />
Group actions, orbits, and stabilizers.<br />
<br />
Orbit-stabilizer theorem.<br />
<br />
Cayley's theorem.<br />
<br />
Cauchy's theorem.<br />
<br />
'''WEEK 6'''<br />
<br />
Applications of orbit-stabilizer theorem (p-groups have nontrivial center, first Sylow theorem.)<br />
<br />
Classification of finite abelian groups and finitely generated abelian groups.<br />
<br />
Composition series and the Jordan-Holder theorem (which we state but don't prove.) <br />
<br />
The difference between knowing the composition factors and knowing the group (e.g. all p-groups of the same order have the same composition factors.)<br />
<br />
'''WEEK 7'''<br />
<br />
Simplicity of A_n.<br />
<br />
Nilpotent groups (main example: the Heisenberg group)<br />
<br />
Derived series and lower central series.<br />
<br />
Category theory 101: Definition of category and functor. Some examples. A group is a groupoid with one object.<br />
<br />
'''WEEK 8'''<br />
<br />
Introduction to representation theory.<br />
<br />
'''WEEK 10'''<br />
<br />
Ring theory 101: Rings, ring homomorphisms, ideals, isomorphism theorems. Examples: fields, Z, the Hamilton quaternions, matrix rings, rings of polynomials and formal power series, quadratic integer rings, group rings. Integral domains. Maximal and prime ideals. The nilradical.<br />
<br />
Module theory 101: Modules, module homomorphisms, submodules, isomorphism theorems. Noetherian modules and Zorn's lemma. Direct sums and direct products of arbitrary collections of modules.<br />
<br />
<br />
==HOMEWORK 1 (due Sep 16)==<br />
<br />
1. Suppose that H_1 and H_2 are subgroups of a group G. Prove that the intersection of H_1 and H_2 is a subgroup of G.<br />
<br />
2. Recall that S_3 (the symmetric group) is the group of permutations of the set {1..3}. List all the subgroups of S_3.<br />
<br />
3. We can define an equivalence relation on rational numbers by declaring two rational numbers to be equal whenever they differ by an integer. We denote the set of equivalence classes by Q/Z. The operation of addition makes Q/Z into a group.<br />
<br />
a) For each n, prove that Q/Z has a subgroup of order n.<br />
<br />
b) Prove that Q/Z is a ''divisible'' group: that is, if x is an element of Q/Z and n is an integer, there exists an element y of Q/Z such that ny = x. (Note that we write the operation in this group as addition rather than multiplication, which is why we write ny for the n-fold product of y with itself rather than y^n)<br />
<br />
c) Prove that Q/Z is not finitely generated. (Hint: prove that if x_1, .. x_d is a finite subset of Q/Z, the subgroup of Q/Z generated by x_1, ... x_d is finite.)<br />
<br />
d) Conclude that Q is not finitely generated.<br />
<br />
4. We will prove that there is no homomorphism from SL_2(Z) to Z except the one which sends all of SL_2(Z) to 0. Suppose f is a homomorphism from SL_2(Z) to Z.<br />
<br />
a) Let U1 be the upper triangular matrix with 1's on the diagonal and a 1 in the upper right hand corner, as in class, and let U2 be the transpose of U1, also as in class. Show that (U1 U2^{-1})^6 = identity (JING 1, TAO 0) and explain why this implies that f(U1) = f(U2).<br />
<br />
b) Show that there is a matrix A in SL_2(Z) such that A U1 A^{-1} = U2^{-1}. (Recall that we say U1 and U2^{-1} are "conjugate".) Explain why this also implies that f(U1) = -f(U2).<br />
<br />
c) Explain why a) and b) imply that f must be identically 0.<br />
<br />
5. The argument above also shows that there is no nonzero homomorphism from SL_2(Z) to Z/pZ where p is a prime greater than 3. However, it leaves open the possibility that there is indeed a nonzero homomorphism from SL_2(Z) to Z/2Z. Exhibit such a homomorphism. Optional challenge problem: exhibit a nonzero homomorphism from SL_2(Z) to Z/3Z.<br />
<br />
<br />
==HOMEWORK 2 (due Sep 25)==<br />
<br />
1. Let V be the Klein 4-group in S_4. Let Q be the symmetric group on the set DF of double-flips in S_4; there are 3 double flips, so Q is isomorphic to S_3. If g is an element of S_4, we discussed in class that conjugation by g permutes the elements of DF. So to each element of g, we have associated an element f(g) of Q. More or less by definition, this defines a homomorphism f: G -> Q. Show that this homomorphism is surjective and has kernel H, and thus that G/H is isomorphic to Q.<br />
<br />
2. Let F_2 be the free group on two generators, which we denote x,y. Prove that there is a automorphism of F_2 which sends x to xyx and y to xy, and prove that this automorphism is unique.<br />
<br />
3. Let H be a subgroup of G, and let N_G(H), the ''normalizer'' of H in G, be the set of elements g in G satisfying g H g^{-1} = H. Prove that N_G(H) is a subgroup of G.<br />
<br />
4. Let T be the subgroup of GL_2(R) consisting of diagonal matrices. This is an example of a "Cartan subgroup," or "torus" (whence the notation T.) Describe the normalizer N(T) of T in GL_2(R). Prove that there is a homomorphism from N(T) to Z/2Z whose kernel is precisely T. <br />
<br />
5. Prove that the group of inner automorphisms of a group is normal in the full automorphism group Aut(G).<br />
<br />
6. Let H and H' be subgroups of a group G, and let x be an element of G. We denote by HxH' the set of elements of the form hxh', with h in H and h' in H', and we call HxH a ''double coset'' of the pair (H,H').<br />
<br />
a) Show that G decomposes as a disjoint union of double cosets of (H,H').<br />
<br />
b) Let G be GL_2(R) and let B be the subgroup of upper triangular matrices. Prove that G decomposes into exactly two double cosets of (B,B). (This is an example of the so-called *Bruhat decomposition* which is of great importance in the theory of algebraic groups and their representations.)<br />
<br />
c) Let G be the symmetric group S_n, and let H be the subgroup of permutations which fix 1. Describe the double coset decomposition of S_n into double cosets of (H,H).<br />
<br />
<br />
==HOMEWORK 3 (due Oct 2)==<br />
<br />
1. Compute the centralizer of the triple flip (12)(34)(56) in S_6. (Note: this is the kind of problem that a computer algebra package like sage is very good at, and I encourage you to learn to use sage well enough to check your answer.)<br />
<br />
2. Warning: a normal subgroup of a normal subgroup need not be normal! Let H be the subgroup of S_4 generated by (12)(34). Then H is a subgroup of the Klein 4-group V_4, which we have already shown is normal in S_4. Show that H is normal in V_4, but H is not normal in S_4.<br />
<br />
3. Let G be a group and let H_1 and H_2 be subgroups of G of finite index (that is, the number of cosets in G/H_i is finite, i = 1,2.) Prove that H_1 intersect H_2 also has finite index in G.<br />
<br />
4. Let X be the set of lines in F_3^2 (here a line means a 1-dimensional linear subspace) so that |X| = 4. Let G be the projective linear group PGL_2(F_3), so that G acts on X via its action on F_3^2. This action can be thought of as a homomorphism from G to S_4. Show that this homomorphism is an ISOMORPHISM from PGL_2(F_3) to S_4.<br />
<br />
5. We proved two theorems in class: Lagrange's theorem, which says that the order of a subgroup of G divides |G|, and Cauchy's theorem, which says that if p divides |G| then there exists a subgroup of G of order p. Cauchy's theorem is a kind of partial converse to Lagrange's theorem, but the full converse is false. Give an example of a finite group G and a divisor n of |G| such that you can prove there is no subgroup of G of order n.<br />
<br />
6. The group S_3 has only three conjugacy classes. Prove that a finite group with at most three conjugacy classes has order at most 6.<br />
<br />
7. Let X be the set of ordered triples of elements of {1,..,n}, for some n >= 3. Then S_n acts on X. How many orbits are there? (Hint: the answer does not depend on n.) OPTIONAL: How many orbits are there on the space of ordered k-tuples, when n >= k? <br />
<br />
==HOMEWORK 4 (due Oct 9)==<br />
<br />
1. Recall that the commutator [x,y] is x y x^{-1} y^{-1}. If G is a group, the commutator subgroup of G, denoted G' or [G,G], is the subgroup of G generated by all commutators [x,y] for x,y in G.<br />
<br />
1a. Show that G' is a normal subgroup of G.<br />
1b. Show that G/G' is an abelian group, which we call the abelianization, G^ab.<br />
1c. Show that if f: G -> A is a homomorphism, with A abelian, then G' is contained in ker f. Conclude that f factors through a homomorphism G/G' -> A.<br />
1d. Show that G' = G if and only if G has no nontrivial abelian quotient. In this case, we say G is ''perfect''.<br />
<br />
2. Prove that any p-Sylow subgroup of GL_n(F_p) is contained in SL_n(F_p).<br />
<br />
3. Let U_n(F_p) be the subgroup of GL_n(F_p) consisting of upper triangular matrices with 1's on the diagonal. Show that U_n(F_p) is a p-Sylow subgroup of GL_n(F_p).<br />
<br />
4. U_n(F_p) is called the *unipotent subgroup* of GL_n(F_p), because it has the property (not part of your homework, but easy to check) that every u in U_n(F_p) is unipotent, i.e. its characteristic polynomial is (x-1)^n. It is a p-group (as you know by virtue of the last problem) so it must have a nontrivial center. What is the center of U_nF_p)?<br />
<br />
5. U_3(F_p) is Shamgar Gurevich's favorite group, the *Heisenberg group" of order p^3. Write down a composition series for U_3(F_p).<br />
<br />
6. Let A be the subgroup of Z^2 generated by the vector (1,0). Let B be the subgroup of Z^2 generated by the vector (3,3). Show that A and B are isomorphic to each other, but Z^2 / A is not isomorphic to Z^2 / B.<br />
<br />
7. Suppose that G and H are groups and f: G -> H is a homomorphism. Recall the definition of the abelianization G^ab from problem 1.<br />
<br />
7a. Show that the composition G -> H -> H^ab factors through a unique homomorphism G^ab -> H^ab, which we denote f^ab.<br />
7b. Show that if f: G -> H and g: H -> Q are homomorphisms, then f^ab o g^ab = (f o g)^ab.<br />
<br />
(For those reading MacLane, this constitutes a proof that abelianization is a '''functor''' from the category of groups to the category of abelian groups.)<br />
<br />
8. Give two different composition series for S_4 and show that they have the same composition factors.<br />
<br />
OPTIONAL: We saw in class that a finitely generated abelian group that was torsion-free was in fact a free abelian group. This totally dies without the hypothesis of abelianness. To see this, give an example of a finitely generated group which is torsion free but which is not a free group, and which does not even *contain* any free group of rank greater than 1.<br />
<br />
==HOMEWORK 5 (due Oct 16)==<br />
<br />
1. Let G be a group, thought of as a category with one object. Show that a set with an action of G is the same thing as a functor from G to the category of sets.<br />
<br />
2. If (V, rho) is a representation of a group G, denote by V^G the space of vectors in V such that gv = v for all g in G; these are called _invariant_ vectors. We saw that the permutation representation of S_3 has a one-dimensional space of invariants. Prove that if X is a finite set with G-action, and V_X the corresponding permutation representation of G, then<br />
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dim V_X^G = number of orbits of X.<br />
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3. (Invariants are functorial) Let G be a group. Then there is a category called C[G]-Mod whose objects are complex representations of G (i.e. pairs (V,rho) where V is a complex vector space and rho is a homomorphism from G to GL(V)). The morphisms in this category are the ones described in class: Mor((V,rho),(W,psi)) is the set of linear maps f from V to W such that<br />
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f(rho(g)(v)) = psi(g)(f(v))<br />
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for all g in G and all v in V.<br />
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Show that there is a functor H_0 from the category of representations of G to the category of vector spaces whose action on objects is given by<br />
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H_0((V) = V^G.<br />
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(In other words: your job again is to explain how to associate to a morphism from (V,rho) to (W,phi) a morphism from V^G to W^G, in a way that's compatible with composition.)<br />
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4. Let G be a finite group, let X be a set with G-action, and let chi be the character of the corresponding permutation representation. Show that, for each g in G, chi(g) is the number of elements of X fixed by g. (I am actually going to prove this in class.)<br />
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5. Let V be the space of homogeneous degree-3 polynomials in variables x_1, x_2, x_3; then S_3 acts on V by permutation of the three variables. Describe the decomposition of V into irreducible representations of S_3.<br />
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6a. Show that if chi is the character of a representation of a finite group G, then chi(g^{-1}) is the complex conjugate of chi.<br />
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6b. As a consequence, show that if g is conjugate to g^{-1} for all g in G, then chi(g) is real.<br />
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6c. The groups D_p and S_n have the property that g is conjugate to g^{-1} for _every_ g in G, so that every character is a real-valued function on G. Give an example of a group G where this is not the case, and give an example of a representation of G whose character takes non-real values.<br />
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OPTIONAL: (for people who know some topology.) Let $X$ be a topological space and let $Paths/X$ be the category of paths up to isotopy as described in class. Explain how to construct a bijection between {functors from Paths/X to the category FinSet of finite sets} and {finite covering spaces of X}. (For simplicity you may assume X is path-connected.)<br />
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==HOMEWORK 6 (due Oct 23)==<br />
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1. Let V_k be the space of homogenous degree-k polynomials in x_1, x_2, x_3. Compute the character of S_3 acting on V_k, and, using this, give formulas for the multiplicity of the trivial rep, the sign rep, and the standard rep of S_3 in V_k.<br />
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2. (Schur's lemma can be false over non-algebraically closed fields) Let G be Z/3Z and let g be a generator of G. Then G acts by cyclic permutations on the set {1,2,3}; let P be the corresponding permutation representation on the REAL (not complex!) vector space generated by e_1,e_2,e_3.<br />
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2a. Show that P breaks up as the direct sum of the trivial representation and a 2-dimensional representation V which is irreducible. (Remember, we are not working over the complex numbers, so you cannot use the criterion for irreducibility in terms of the character that we worked out in class for the complex case; you need to prove directly that V has no nontrivial proper subrepresentation.)<br />
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2b. Show that the map from V to V induced by g is G-equivariant, but is neither zero nor a scalar multiplication.<br />
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The following two problems involve the tensor product. I am going to teach the general notion of tensor product of modules in this course, but I am expecting you already know what the tensor product of vector spaces. If not, please review it by learning the definition and doing these problems, because having a familiarity with the vector space theory will be very helpful in learning the general story.<br />
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3. Let V and W be complex vector spaces. Let f be a linear transformation of V, and g a linear transformation of W. <br />
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3a. Show that there is a unique linear transformation F satisfying<br />
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F(v tensor w) = f(v) tensor g(w)<br />
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for all v in V and all w in W. We denote this transformation by f tensor g.<br />
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3b. Suppose V and W are finite dimensional. Show that Trace(f tensor g) = Trace(f) Trace(g). What does this say when f and g are both the identity transformation?<br />
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4. Let V be a vector space. We define Sym^2 V to be the quotient of V tensor V by the subspace generated by all elements of the form<br />
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v tensor w - w tensor v<br />
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for v,w in V.<br />
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Suppose dim V = n. What is dim Sym^2 V?<br />
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5. For any n, the group S_n has an (n-1)-dimensional representation, the "standard representation," on the space of vectors in C^n whose coordinates sum to 0. When n=3, we proved in class that this representation was irreducible. Prove that the standard representation is irreducible for all n.<br />
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6. Give an example showing that the standard representation of S_n need NOT be irreducible over a field of characteristic p.<br />
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7. Let H be a subgroup of G and let V be a representation of G. Let chi_{G/H} be the character of the permutation representation of G on the set of cosets G/H, so that, by problem 1, chi_{G/H}(g) is the number of fixed points of g in its action on the cosets. As discussed in class, the dimension of the G-invariant subspace V^G is given by the inner product of chi_V with the trivial character. Prove that the dimension of V^H is given by the inner product of chi_V with chi_{G/H}.<br />
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==HOMEWORK 7 (due Oct 30)==<br />
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1. When a formula always yields a non-negative integer, it is often because it is secretly a formula for either the cardinality of some finite set or the dimension of some finite-dimensional vector space. In this case, show that<br />
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<chi_{V_1}, chi_{V_2}> = dim_C Hom_G(V_1,V_2)<br />
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where we recall that Hom_G(V_1,G_2) refers to the space of G-equivariant homomorphisms from V_1 to V_2.<br />
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2. Let G be the group of affine linear transformations of F_5; that is, it is the group of transformations x -> ax + b where a lies in (Z/5Z)^* and b lies in (Z/5Z). Note that G has order 20. Write H for the subgroup of G consisting of transformations fixing 0 (i.e. those for which b=0). <br />
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We will work out all the irreducible representations of G. <br />
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2a. First of all, consider the 5-dimensional permutation representation of G acting by left multiplication on the cosets of H. Show that this representation decomposes as a direct sum of the trivial representation and an irreducible 4-dimensional representation. (Hint: it is probably easier to use the character of this representation than to prove irreducibility directly.) <br />
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2b. Now show that there are 4 1-dimensional representations of G.<br />
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2c. Show that the irreducible representations of G you have constructed are the only ones.<br />
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3. If G and H are finite groups, and V is a representation of G, and W is a representation of H, then V tensor W is a representation of GxH, and the character chi_{V tensor W} is given by<br />
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chi_{V tensor W}(g,h) = chi_V(g) chi_W(h).<br />
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3a. Prove that V tensor W is irreducible if and only if V and W are irreducible.<br />
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3b. Prove that all the irreducible representations of G x H are of the form V tensor W.<br />
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(Together, these propositions show that you can completely describe the representation theory of G x H in terms of that of G and that of H.)<br />
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4. Suppose that V is an irreducible representation of S_n, and suppose that when we consider V as a representation of the alternating group A_n it is NOT irreducible. Prove that V, considered as a representation of A_n, is the direct sum of two non-isomorphic irreducible representations. Prove furthermore that chi_V(g) = 0 for all odd permutations g. Give an example of such an irreducible representation of S_4.<br />
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OPTIONAL (because I couldn't quickly think of an easy way to do it!) Prove that there exists such a representation of S_n for every n >= 3.<br />
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5. (Fourier analysis over finite fields.) I said in class that the representation theory that goes into Fourier analysis is different from the representation theory of finite groups we do in class, but that's not quite true; when you do Fourier theory over finite fields, the two theories come into much closer contact, with the bonus that we don't have to worry about issues of infinite sums that I hand-waved away in class.<br />
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Let F be the field of p elements and let V be the space of complex-valued functions on F; so V is a p-dimensional space. Let G be the group Z/pZ (in other words, it is the additive group of the field.)<br />
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Now V is a representation of G, in the same way we discussed in class: if f is a function in V, and a is an element of g, then the function gf is defined by<br />
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gf(x) = f(x+a)<br />
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5a. Describe the breakup of V into irreducible representations of G, which are all 1-dimensional (not just what the characters are, explicitly decribe the irreducible constituents of V as subspaces of V!)<br />
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5b. There is a natural norm on V which sends f to ||f|| = sum_x |f(x)|^2.<br />
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Given any f, we have a unique decomposition f = sum_i f_i, where f_i lies in the irreducible constituent V_i of V.<br />
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Prove that ||f|| = sum_i ||f_i||.<br />
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OPTIONAL (for people who are taking analysis) Explain why ||f_i|| is correctly thought of as a "Fourier coefficient" and why the fact proved in 5b is the finite-field analogue of Parseval's identity.<br />
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==HOMEWORK 8 (due Nov 6)==<br />
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1. If R is a ring (not necessarily with 1) such that a^2 = a for every a in R, R is called a Boolean ring.<br />
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1a. Show that any nonzero Boolean ring has characteristic 2. (In other words, show that a+a=0 for all a in R.)<br />
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1b. Show that every Boolean ring is commutative.<br />
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2. Let phi : R -> S be a homomorphism of commutative rings (recalling that rings for us have a multiplicative identity 1 and that ring homomorphisms take 1 to 1.)<br />
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2a. If I is an ideal of S, show that phi^(-1) (I) is an ideal of R.<br />
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2b. If P is a prime ideal of S, show that phi^(-1) (P) is either R or a prime ideal of R.<br />
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2c. If M is a maximal ideal of S and phi is surjective, show that phi^(-1) (M) is a maximal ideal of R.<br />
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2d. Find a homomorphism phi : R -> S and a maximal ideal M of S such that phi^(-1) (M) is not a maximal ideal of R.<br />
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3. Let Nil(R) denote the set of nilpotent elements of a ring R; that is, the set of x such that x^k = 0 for some positive integer k. <br />
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3a. Show that Nil(R) is an ideal whenever R is a commutative ring. Assume R is commutative for the remainder of this exercise.<br />
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3b. Describe Nil(Z/720Z).<br />
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3c. If x is in Nil(R) and u is a unit, show that u+x is a unit.<br />
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3d. Show that Nil(M_2(R)) is not an ideal of the matrix algebra M_2(R). (That is, it is neither a left nor a right ideal.)<br />
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4. We say a ring is Noetherian if any ascending chain of ideals I_1 < I_2 < I_3 < …. eventually stabilizes. This is a somewhat funny-looking condition if you haven't encountered it before, but it turns out to be a very useful way of formalizing the notion that a ring is "not too bad."<br />
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4a. Show that Z is Noetherian.<br />
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4b. Let R be the ring of continuous functions from the real numbers to the real numbers. Show that R is not Noetherian.<br />
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5. '''How algebraists do calculus'''. The ''ring of dual numbers'' is the ring A = C[e]/(e^2). We think of e as an infinitesimal number; that is, something so small that its square is 0. This algebraic notion of infinitesimal is the basis for the algebraic notion of differentiation, which works as follows. Let f be a polynomial in C[x]. Show that there is a unique polynomial g in C[x] such that<br />
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f(x+e) - f(x) = eg(x)<br />
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and that in fact g is the derivative of f. In other words, the usual definition of "derivative" works just fine in this context, without any use of the notion of limit!<br />
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6. Write down all the ideals of C[e]/e^2. (There are three.)<br />
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7. Show that any right ideal of the ring M_2(Q) of 2x2 matrices is an ideal of the form I_V described in class for some subspace V of Q^2; that is, it consists precisely of those linear transformations whose image is contained in V.<br />
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== HOMEWORK 9 (Due Nov 13) ==<br />
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1. If M is an R-module and I is an ideal of R such that im = 0 for all i in I and m in M, show that M carries the structure of an R/I-module.<br />
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2. It follows from 1. that if S is a commutative ring and J is a maximal ideal of S, that J/J^2 is a vector space over the field S/J. Compute the dimension of J/J^2 as a vector space over S/J when<br />
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2a. S = Z, J = (5);<br />
2b. S = C[x,y], J = (x,y)<br />
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2c. Suppose S = C[x,y]/(f) and J = (x,y). Then the dimension of J/J^2 depends on the choice of the polynomial f (where, to ensure that J/(f) makes sense, you should assume that f is in J). Compute the possible values of this dimension and give an example of an f realizing each possibility. (NOTE: this is secretly, or not-so-secretly, another instance of differential geometry as carried out by algebraists...)<br />
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3. An ''idempotent'' element of a ring is an element e satisfying e^2 = e.<br />
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3a. If e is an idempotent, show that 1-e is also an idempotent.<br />
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3b. Suppose R has a unit (we are actually always supposing this, but I just want to remind you of it for this problem) and suppose e is a ''central'' idempotent, i.e. an idempotent which commutes with all elements of R. Show that eR and (1-e)R are both two-sided ideals. Moreover, show that eR and (1-e)R are also subrings of R (with the warning that the identity in eR is not 1, which is not contained in eR, but rather e.)<br />
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3c. Show that the map f_e: R -> eR defined by f_e(r) = er is a ring homomorphism. Finally show that the ring homomorphism<br />
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f_{e} x f_{1-e}: R -> eR x (1-e)R <br />
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is actually an isomorphism of rings. In other words, the presence of a central idempotent induces a decomposition a ring as a direct product.<br />
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4. Let k be an algebraically closed field, and let D be a finite-dimensional division algebra over k. Prove that D = k. (Hint: if alpha is an element of D which is not in k, prove that there is some polynomial f(x) such that f(alpha) = 0. Then think back to our proof in class that the quaternion algebra with complex coefficients was not a division algebra.)<br />
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5. The ''center'' of a ring R is the subring of elements commuting with every element of R.<br />
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5a. Show that the center of M_n(Q) is the ring of scalar matrices.<br />
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5b. Show that the center of the group ring C[G] is the set of elements sum f(g) g such that f(g) is a class function. Compute the central idempotents of C[S_3].<br />
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6. '''How algebraists do differential equations.''' We define the ''Weyl algebra'' W of differential operators in one variable as follows. Multiplication by x is a linear transformation on the complex vector space V of all smooth functions on C. Differentiation, which we denote d, is also a linear transformation on V. So we let W be the subring of End(V) generated by x and d. We note that x and d satisfy the relation xd = dx - 1; in particular, the Weyl algebra is not commutative.<br />
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6a. Describe the submodule of V which is annihilated by the element d-x in W.<br />
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6b. Show that, for any polynomial f(x), fd-df = -f', and also show that for any polynomial g(d), gx-xg = g'. (This is another way algebraists do calculus!) <br />
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6c. Use the facts from 6b. to show that the center of the Weyl algebra is just the field of constants. (Remark: you will use in the course of this proof the fact that a polynomial whose derivative is 0 is constant -- this would not be true if we were working in positive characteristic, as the example of x^p considered as a polynomial over F_p demonstrates.)<br />
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7. If M is a left module for a ring R, we denote by Ann(M) the set of elements r in R such that rm = 0 for all m in M. Show that Ann(M) is a two-sided ideal. <br />
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8. Let M be a left module for R, and let m be an element of M. We denote by Ann(m) the set of elements r in R such that rm = 0.<br />
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8a. Show that Ann(m) is a left ideal, and show that the submodule Rm of M generated by m is isomorphic to the left module R/Ann(m).<br />
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8c. Let V,W be as in question 6. Show that Ann(e^x) is the left ideal W(d-1).<br />
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==HOMEWORK 10 (due Nov 22)==<br />
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1. Let f: R -> S be a homomorphism of rings. Then S naturally has the structure of R-bimodule by left and right multiplication (technically, r acts by multiplication by f(r).) If M is a left R-module, then S tensor_R M is a left S-module. On the other hand, if N is an S-module, then N can also be considered as an R-module by pullback of the action. These two operations are functors: the first, a functor from left R-modules to left S-modules, the second, a functor from left S-modules to left R-modules.<br />
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Prove that if M is a finitely generated left R-module, then S tensor_R M is a finitely generated left S-module. On the other hand, give an example to show that N can be finitely generated as a left S-module but not finitely generated when considered as a left R-module.<br />
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2. Prove a fact I stated in class: if A and B are abelian groups (i.e. Z-modules) which are finite, and |A| and |B| are relatively prime, then A tensor_Z B = 0.<br />
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3. Prove the converse: if A and B are finite abelian groups and A tensor_Z B = 0, then |A| and |B| are relatively prime.<br />
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4. Let R be a commutative ring with 1. If R has a unique maximal ideal M, R is called a "local ring".<br />
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4a. If R is a local ring, show that every element of R that is not in M is a unit. (I think Rob H. proved this in class.)<br />
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4b. Conversely, if R is such that the set of nonunits forms an ideal I, show that R is a local ring with maximal ideal I.<br />
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5a. If R is a ring, we denote R^op ("the opposite ring") to be the ring whose elements are those of R, but whose multiplication law x_opp is defined by s x_opp r = rs. Show that if M is a left R-module, then the map M x R^op -> M sending (x,r) to rx makes M into a right R^op module. In particular, if there is a RING ISOMORPHISM from R to R^op, then every left R-module can be thought of as a right R-module.<br />
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5b. Verify that the map sending g to g^{-1} extends by linearity to give a ring isomorphism between C[G] and C[G]^op.<br />
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5c. Construct a ring isomorphism between M_n(Q) and M_n(Q)^op.<br />
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5d. Optional: Give an example of a ring (with 1) which is *not* isomorphic to its opposite ring. (This is not so easy, I think! I found a MathOverflow thread with several examples, none of which came with a short proof of the non-isomorphism -- try your hand!)<br />
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6. Let f: C[x] tensor_C C[y] -> C[z] be the unique map satisfying f(x^a tensor y^b) = z^{a+b}. Show that f(m tensor n) is nonzero for any nonzero pure tensor m tensor n, but f is not injective. (This is a warning not to try to check injectivity by checking that no pure tensors are killed by the map.)<br />
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7. Let R be a commutative local ring with maximal ideal m, and A -> B a map of R-modules. <br />
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7a. Show that if A -> B is surjective, than so is A tensor R/m -> B tensor R/m. (In fact, this has nothing to do with R being local.)<br />
7b. Show that if B is a finitely generated R-module, the CONVERSE is true: if A tensor R/m -> B tensor R/m is surjective, then so is A -> B (use Nakayama)<br />
7c. Show that the converse is not true without the hypothesis of finite generation. For instance, let R = C[[t]], and give an example of a map A -> B of R-modules which is not surjective, but such that A/tA -> B/tA is surjective.<br />
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<!--Below you will find a repository of homework problems. Note that some of these problems are taken from Lang's ''Algebra''.<br />
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==HOMEWORK 1 (due Sep 20)==<br />
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1. Suppose that H_1 and H_2 are subgroups of a group G. Prove that the intersection of H_1 and H_2 is a subgroup of G.<br />
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2. Recall that S_3 (the symmetric group) is the group of permutations of the set {1..3}. List all the subgroups of S_3.<br />
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3. We can define an equivalence relation on rational numbers by declaring two rational numbers to be equal whenever they differ by an integer. We denote the set of equivalence classes by Q/Z. The operation of addition makes Q/Z into a group.<br />
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a) For each n, prove that Q/Z has a subgroup of order n.<br />
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b) Prove that Q/Z is a ''divisible'' group: that is, if x is an element of Q/Z and n is an integer, there exists an element y of Q/Z such that ny = x. (Note that we write the operation in this group as addition rather than multiplication, which is why we write ny for the n-fold product of y with itself rather than y^n)<br />
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c) Prove that Q/Z is not finitely generated. (Hint: prove that if x_1, .. x_d is a finite subset of Q/Z, the subgroup of Q/Z generated by x_1, ... x_d is finite.)<br />
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4. Let F_2 be the free group on two generators x,y. Prove that there is a automorphism of F_2 which sends x to xyx and y to xy, and prove that this automorphism is unique.<br />
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5. Let H be a subgroup of G, and let N_G(H), the ''normalizer'' of H in G, be the set of elements g in G satsifying g H g^{-1} = H. Prove that N_G(H) is a subgroup of G.<br />
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6. Let T be the subgroup of GL_2(R) consisting of diagonal matrices. This is an example of a "Cartan subgroup," or "torus" (whence the notation T.) Describe the normalizer N(T) of T in GL_2(R). Prove that there is a homomorphism from N(T) to Z/2Z whose kernel is precisely T. <br />
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7. Prove that the group of inner automorphisms of a group is normal in the full automorphism group Aut(G).<br />
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8. Let H and H' be subgroups of a group G, and let x be an element of G. We denote by HxH' the set of elements of the form hxh', with h in H and h' in H', and we call HxH a ''double coset'' of the pair (H,H').<br />
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a) Show that G decomposes as a disjoint union of double cosets of (H,H').<br />
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b) Let G be GL_2(R) and let B be the subgroup of upper triangular matrices. Prove that G decomposes into exactly two double cosets of (B,B).<br />
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c) Let G be the symmetric group S_n, and let H be the subgroup of permutations which fix 1. Describe the double coset decomposition of S_n into double cosets of (H,H).<br />
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==HOMEWORK 2 (due Sep 27)==<br />
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1. Prove that the symmetric group S_n is generated by transpositions. Hint: proceed by induction on n. Let H be the subgroup of S_n consisting of permutations fixing 1. Observe that H is isomorphic to S_{n-1}, so that H is generated by transpositions. Explain why it suffices to show that, for any g in S_n, the coset gH contains a transposition. Then explain why this is the case.<br />
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2. Given that the symmetric group S_n is generated by transpositions, show that there are no nontrivial homomorphisms from S_n to Z/pZ for any odd prime p.<br />
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3. Compute the centralizer of the triple flip (12)(34)(56) in S_6. (Note: this is the kind of problem that a computer algebra package like sage is very good at, and I encourage you to learn to use sage well enough to check your answer.)<br />
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4. Warning: a normal subgroup of a normal subgroup need not be normal! Let H be the subgroup of S_4 generated by (12)(34). Then H is a subgroup of the Klein 4-group V_4, which we have already shown is normal in S_4. Show that H is normal in V_4, but H is not normal in S_4.<br />
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5. The '''quaternion group''' Q is a non-abelian finite group of order 8. Its elements are 1, -1, i, -i, j, -j, k, -k; it satisfies g^2 = -1 for all g except +-1, and ijk = 1. (Note that the notation "-i" means "the product of i with -1" and -1 is understood to be central in Q.) Prove that every subgroup of Q is normal.<br />
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6. (finishing example done in class) Let Gamma be the group F<x,y> / (x^2 = 1, y^2 = 1). (Remember, this means the quotient of the free group by the group generated by all conjugates of x^2 and y^2.) Prove that Gamma is infinite. Prove that Gamma has a subgroup R generated by xy isomorphic to Z, that R is normal, and that the quotient Gamma/R is isomorphic to Z/2Z. <br />
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7. Suppose a group G has the property that every element in g satisfies g^2 = 1. Show that G must be abelian.<br />
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==HOMEWORK 3 (due Oct 4)==<br />
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1. Multiplication by -1 is an automorphism of Z/nZ. Let a be the homomorphism Z/2Z -> Aut(Z/nZ) which sends the nontrivial element of Z/2Z to multiplication by -1. Then we have a semidirect product of Z/pZ by Z/2Z as defined in class. This group of order 2n is called a ''dihedral group.'' and is denoted D_n (or sometimes D_{2n}).<br />
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1a. Compute the center of D_n. (Note that the answer depends on n!)<br />
1b. Show that all involutions of D_n are conjugate to each other if and only if n is odd.<br />
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2. (More on direct products that we didn't do in class.) Suppose that G is a semidirect product of N with H. Suppose furthermore that H is also normal in G. Prove that G is a direct product of N with H.<br />
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3. Let Q be the quaternion group from last week's homework. Show that Q does not decompose as a semidirect product N \rtimes H, apart form the trivial cases N = {e} and N = Q.<br />
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4. The ''affine linear group'' of degree n is the group of transformations from R^n to R^n of the form x -> Ax + b, where A is a matrix in GL_n(R) and b is a vector in R^n. Prove that the affine linear group is a semidirect product R^n \rtimes GL_n.<br />
<br />
5. The ''ordinary triangle group'' T(p,q,r) is the group with presentation <x,y | x^p = y^q = (xy)^r = 1>. This is a very interesting family of groups which arises naturally in many contexts in number theory and topology.<br />
<br />
6a. Prove that T(2,2,n) is isomorphic to the dihedral group D_n.<br />
6b. Prove that T(2,3,3) is finite and compute its order. Is it isomorphic to a finite group we've discussed in class?<br />
<br />
(It is a beautiful fact, but outside the scope of this course, that T(p,q,r) is finite if and only if 1/p + 1/q + 1/r > 1.)<br />
<br />
7. Let G be a subgroup of GL_n(R) which contains SL_n(R). Prove that G is a normal subgroup of GL_n(R).<br />
<br />
8. Let G be a group and let H_1 and H_2 be subgroups of G of finite index (that is, the number of cosets in G/H_i is finite, i = 1,2.) Prove that H_1 intersect H_2 also has finite index in G.<br />
<br />
9. Recall that the commutator [x,y] is x y x^{-1} y^{-1}. If G is a group, the commutator subgroup of G, denoted G' or [G,G], is the subgroup of G generated by all commutators [x,y] for x,y in G.<br />
<br />
9a. Show that G' is a normal subgroup of G.<br />
9b. Show that G/G' is an abelian group.<br />
9c. Show that if f: G -> A is a homomorphism, with A abelian, then G' is contained in ker f. Conclude that f factors through a homomorphism G/G' -> A.<br />
9d. Show that G' = G if and only if G has no nontrivial abelian quotient. In this case, we say G is ''perfect''.<br />
<br />
10. Under what conditions on (p,q,r) is the triangle group T(p,q,r) perfect?<br />
<br />
==HOMEWORK 4 (due Oct 16)==<br />
<br />
1. Let X be a set with a transitive action of a group G, which we think of as a homomorphism f: G -> Sym(X). Let H_x be the stabilizer of an element x of X.<br />
<br />
1a. If x' is another element of X, show that H_{x"} and H_x are conjugate subgroups. (Hint: this is not true without the hypothesis of transitivity, so make sure to use it!)<br />
<br />
1b. Show that the kernel of f is the intersection of all the conjugates of H_x in G. (This subgroup is called the ''normal core'' of H_x.)<br />
<br />
2. This week, a certified mathematical grown-up asked on MathOverflow what the index-3 subgroups of SL_2(Z/pZ) were. Let's do it ourselves for homework. Let p be a prime, and suppose H is an index-3 subgroup of SL_2(Z/pZ). I do not claim that the proof sketched below is necessarily the easiest, but it allows us to cover some ground that we won't get to do in lecture.<br />
<br />
2a. Using the coset action, show that the normal core of H is the kernel of a homomorphism f from SL_2(Z/pZ) to S_3.<br />
<br />
2b. A ''unipotent matrix'' is a matrix A such that (A-I)^n = 0 for some positive integer n. Prove that a unipotent element of SL_2(Z/pZ) has order dividing p.<br />
<br />
2c. You may use the fact that unipotent matrices generate SL_2(Z/pZ). Given this fact, show that f must be trivial when p > 3.<br />
<br />
2d. Using the above, show that SL_2(Z/pZ) has no index-3 subgroups when p > 3.<br />
<br />
2e. (extra credit) What can you say about the minimal index of a proper subgroup of SL_2(Z/pZ)?<br />
<br />
3. Let H be a subgroup of G of index 2. Prove that H is normal.<br />
<br />
4. We proved two theorems in class: Lagrange's theorem, which says that the order of a subgroup of G divides |G|, and Cauchy's theorem, which says that if p divides |G| then there exists a subgroup of G of order p. Cauchy's theorem is a kind of partial converse to Lagrange's theorem, but the full converse is false. Give an example of a finite group G and a divisor n of |G| such that you can prove there is no subgroup of G of order n.<br />
<br />
5. Let X be the set of lines in F_3^2 (here a line means a 1-dimensional linear subspace) so that |X| = 4. Let G be the projective linear group PGL_2(F_3), so that G acts on X via its action on F_3^2. This action can be thought of as a homomorphism from G to S_4. Show that this homomorphism is an ISOMORPHISM from PGL_2(F_3) to S_4.<br />
<br />
6. The group S_3 has only three conjugacy classes. Prove that a finite group with at most three conjugacy classes has order at most 6.<br />
<br />
7. Let X be the set of ordered triples of elements of {1,..,n}, for some n >= 3. Then S_n acts on X. How many orbits are there? (Hint: the answer does not depend on n.)<br />
<br />
8. Suppose that G and H are groups and f: G -> H is a homomorphism. Recall the definition of the abelianization G^ab from the previous problem set.<br />
<br />
8a. Show that the composition G -> H -> H^ab factors through a unique homomorphism G^ab -> H^ab, which we denote f^ab.<br />
<br />
8b. Show that if f: G -> H and g: H -> Q are homomorphisms, then f^ab o g^ab = (f o g)^ab.<br />
<br />
(For those reading MacLane, this constitutes a proof that abelianization is a '''functor''' from the category of groups to the category of abelian groups.)<br />
<br />
==HOMEWORK 5 (due Oct 23)==<br />
<br />
1. Show that there are only two different isomorphism classes of groups of order 14. (Hint: prove that such a group must be a semidirect product of a group of order 7 by a group of order 2.)<br />
<br />
2. (variant of Burnside's Lemma) Let G be a finite group and H a normal subgroup. Suppose G acts on a finite set X. <br />
<br />
2a. Show that if x and y lie in the same H-orbit, and g is an element of G, then gx and gy also lie in the same H-orbit. Thus, G/H has a natural action on the set of H-orbits X/H.<br />
<br />
2b. Show that the average number of fixed points in X of an element of the coset gH is the same as the number of fixed points of the element gH of G/H in its action on X/H.<br />
<br />
3. A ''central extension'' of a group G by an abelian group A is a group E, together with a surjective homomorphism E -> G whose kernel is central in E and is isomorphic to A.<br />
<br />
3a. Show that the quaternion group is a central extension of (Z/2Z)^2 by Z/2Z.<br />
<br />
3b. Give an example of another nonabelian group which is a central extension of (Z/2Z)^2 by Z/2Z and which is not isomorphic to the quaternion group.<br />
<br />
4. Prove that a central extension of an abelian group is nilpotent.<br />
<br />
5. Give two different composition series for S_4 and show that they have the same composition factors.<br />
<br />
6. Suppose that A is a finitely generated abelian group, and B is a subgroup of A.<br />
<br />
6a. Prove that B is finitely generated and that rank(B) <= rank(A).<br />
<br />
6b. Prove that rank(A/B) = rank(A) - rank(B).<br />
<br />
7. For this problem, you may use the fact (which I should have asked on homework earlier) that if H_1 and H_2 are subgroups of G, then<br />
<br />
[G: H_1 intersect H_2] <= [G:H_1][G:H_2].<br />
<br />
Prove that if G is a simple group, d is an integer greater than 1, and S is a subgroup of index d in G, then |G| <= d^d.<br />
<br />
Conclude that, in particular, A_5 has no subgroup of index 3. On the other hand, exhibit a subgroup of A_5 of index 5.<br />
<br />
(Note: d^d is not a sharp bound; can you get a bound of d factorial instead?)<br />
<br />
<!--(This argument shows that a solvable subgroup of A_n has size at most about (n-1)!, but this is far from sharp; e.g. I found a citation to a 1967 paper of Dixon which shows that a solvable subgroup of A_n has size at most k^n where k is around 2.88.)--><br />
<br />
<!--<br />
8. Give an example of an abelian group A with a subgroup B such that B is not a direct summand of A.<br />
<br />
9. Let A be the subgroup of Z^2 generated by the vector (1,0). Let B be the subgroup of Z^2 generated by the vector (3,3). Show that A and B are isomorphic to each other, but Z^2 / A is not isomorphic to Z^2 / B.<br />
<br />
==HOMEWORK 6 (due Oct 30)==<br />
<br />
1. In this problem, you will construct a functor F from the category of finite sets to the category of complex vector spaces. I'll tell you what it does on objects: if X is a finite set, then F(X) is the vector space spanned by a set of basis elements {e_x}_{x in X}, so that dim F(X) = |X|. Your job: complete the definition of F by describing the map from Mor_{Finite Sets}(X,Y) to Mor_{Vector Spaces}(F(X),F(Y)), and showing that your map respects composition.<br />
<br />
2. If (V, rho) is a representation of a group G, denote by V^G the space of vectors in V such that gv = v for all g in G; these are called _invariant_ vectors. We saw that the permutation representation of S_3 has a one-dimensional space of invariants. Prove that if X is a finite set with G-action, and V_X the corresponding permutation representation of G, then<br />
<br />
dim V_X^G = number of orbits of X.<br />
<br />
3. (Invariants are functorial) Let G be a group. Then there is a category called C[G]-Mod whose objects are complex representations of G (i.e. pairs (V,rho) where V is a complex vector space and rho is a homomorphism from G to GL(V)). The morphisms in this category are the ones described in class: Mor((V,rho),(W,psi)) is the set of linear maps f from V to W such that<br />
<br />
f(rho(g)(v)) = psi(g)(f(v))<br />
<br />
for all g in G and all v in V.<br />
<br />
Show that there is a functor H_0 from the category of representations of G to the category of vector spaces whose action on objects is given by<br />
<br />
H_0((V) = V^G.<br />
<br />
(In other words: your job again is to explain how to associate to a morphism from (V,rho) to (W,phi) a morphism from V^G to W^G, in a way that's compatible with composition.)<br />
<br />
<!--4. The group PSL_2(Z) has a presentation of the form [U,V|U^2 = V^3 = 1]. Thus, an n-dimensional representation of PSL_2(Z) is precisely a pair of n x n matrices (rho(U),rho(V)), such that rho(U)^2 = rho(V)^3 = 1. Prove that for every n > 1 there are uncountably many isomorphism classes of n-dimensional representations of PSL_2(Z). (Extra credit: show that there are uncountably many isomorphism classes of _irreducible_ representations.)<br />
<br />
An amazing theorem of Margulis (superrigidity) shows that SL_2(Z) is very special among arithmetic groups in this way. For instance, there are only _finitely_ many isomorphism classes of n-dimensional representations of PSL_d(Z) for any d > 2; those groups are "rigid."--><br />
<br />
<!--<br />
5. Let G be a finite group, let X be a set with G-action, and let chi be the character of the corresponding permutation representation. Show that, for each g in G, chi(g) is the number of elements of X fixed by g. (I am actually going to prove this in class.)<br />
<br />
6. Let V be the space of homogeneous degree-3 polynomials in variables x_1, x_2, x_3; then S_3 acts on V by permutation of the three variables. Describe the decomposition of V into irreducible representations of S_3.<br />
<br />
7a. Show that if chi is the character of a representation of a finite group G, then chi(g^{-1}) is the complex conjugate of chi.<br />
<br />
7b. As a consequence, show that if g is conjugate to g^{-1} for all g in G, then chi(g) is real.<br />
<br />
7c. The groups D_p and S_n have the property that g is conjugate to g^{-1} for _every_ g in G, so that every character is a real-valued function on G. Give an example of a group G where this is not the case, and give an example of a representation of G whose character takes non-real values.<br />
<br />
==HOMEWORK 7 (due Nov 6)==<br />
<br />
1. Let G be a finite group, let X be a finite set with G-action, and let chi be the character of the corresponding permutation representation. Show that, for each g in G, chi(g) is the number of elements of X fixed by g. (I am actually going to prove this in class.)<br />
<br />
2a. Show that if chi is the character of a finite-dimensional representation of a finite group G, then chi(g^{-1}) is the complex conjugate of chi(g) for all g.<br />
<br />
2b. As a consequence, show that if g is conjugate to g^{-1} for all g in G, then chi(g) is real.<br />
<br />
2c. The groups D_p and S_n have the property that g is conjugate to g^{-1} for _every_ g in G, so that every character is a real-valued function on G. Give an example of a group G where this is not the case, and give an example of a representation of G whose character takes non-real values.<br />
<br />
3. Let V and W be complex vector spaces. Let f be a linear transformation of V, and g a linear transformation of W. <br />
<br />
3a. Show that there is a unique linear transformation F satisfying<br />
<br />
F(v tensor w) = f(v) tensor g(w)<br />
<br />
for all v in V and all w in W. We denote this transformation by f tensor g.<br />
<br />
3b. Suppose V and W are finite dimensional. Show that Trace(f tensor g) = Trace(f) Trace(g). What does this say when f and g are both the identity transformation?<br />
<br />
4. Let V be a vector space. We define Sym^2 V to be the quotient of V tensor V by the subspace generated by all elements of the form<br />
<br />
v tensor w - w tensor v<br />
<br />
for v,w in V.<br />
<br />
Suppose dim V = n. What is dim Sym^2 V?<br />
<br />
5. Show that Z/pZ tensor Z/qZ is zero when p and q are distinct primes.<br />
<br />
==HOMEWORK 8 (due Nov 13)==<br />
<br />
1. (Schur's lemma can be false over non-algebraically closed fields) Let G be Z/3Z and let g be a generator of G. Then G acts by cyclic permutations on the set {1,2,3}; let P be the corresponding permutation representation on the REAL (not complex!) vector space generated by e_1,e_2,e_3.<br />
<br />
1a. Show that P breaks up as the direct sum of the trivial representation and a 2-dimensional representation V which is irreducible. (Remember, we are not working over the complex numbers, so you cannot use the criterion for irreducibility in terms of the character that we worked out in class for the complex case; you need to prove directly that V has no nontrivial proper subrepresentation.)<br />
<br />
1b. Show that the map from V to V induced by g is G-equivariant, but is neither zero nor a scalar multiplication.<br />
<br />
2. If V_1 and V_2 are representations (not necessarily irreducible!) of G, show that <chi_V_1, chi V_2> is a non-negative integer.<br />
<br />
3. When a formula always yields a non-negative integer, it is often because it is secretly a formula for either the cardinality of some finite set or the dimension of some finite-dimensional vector space. In this case, show that<br />
<br />
<chi_{V_1}, chi_{V_2}> = dim_C Hom_G(V_1,V_2)<br />
<br />
where we recall that Hom_G(V_1,G_2) refers to the space of G-equivariant homomorphisms from V_1 to V_2.<br />
<br />
4. If A is an abelian group, show that every irreducible representation of A is 1-dimensional. (Hint: show that there is a vector which is an eigenvector for every element of A!)<br />
<br />
5. Let G be the group of order 20 which is the semidirect product of A by H, where A is Z/5Z and H is (Z/5Z)^* with its natural action on A. We will work out all the irreducible representations of A. <br />
<br />
5a. First of all, consider the 5-dimensional permutation representation of G acting by left multiplication on the cosets of H. Show that this representation decomposes as a direct sum of the trivial representation and an irreducible 4-dimensional representation. (Hint: it is probably easier to use the character of this representation than to prove irreducibility directly.) <br />
<br />
5b. Now show that there are 4 1-dimensional representations of G which factor through the quotient G/A (i.e. such that rho(A) is the identity.)<br />
<br />
5c. Using the decomposition of the regular representation, show that the irreducible representations of G you have constructed are the only ones.<br />
<br />
==HOMEWORK 9 (due Nov 20)==<br />
<br />
0. Find all ring homomorphisms from Z to Z/30Z. [Warning: there are 8 of them, not 30.]<br />
<br />
<br />
1. If R is a ring (not necessarily with 1) such that a^2 = a for every a in R, R is called a Boolean ring.<br />
<br />
1a. Show that any nonzero Boolean ring has characteristic 2. (In other words, show that a+a=0 for all a in R.)<br />
<br />
1b. Show that every Boolean ring is commutative.<br />
<br />
<br />
2. Let phi : R -> S be a homomorphism of commutative rings with 1.<br />
<br />
2a. If I is an ideal of S, show that phi^(-1) (I) is an ideal of R.<br />
<br />
2b. If P is a prime ideal of S, show that phi^(-1) (P) is either R or a prime ideal of R.<br />
<br />
2c. If M is a maximal ideal of S and phi is surjective, show that phi^(-1) (M) is a maximal ideal of R.<br />
<br />
2d. Find a homomorphism phi : R -> S and a maximal ideal M of S such that phi^(-1) (M) is not a maximal ideal of R.<br />
<br />
<br />
3. Let Nil(A) denote the set of nilpotent elements of a ring A, and let R be a commutative ring. In class it was observed that Nil(R) is an ideal of R.<br />
<br />
3a. Find (a generator of) Nil(Z/720Z).<br />
<br />
3b. Show that Nil(R/Nil(R)) = 0.<br />
<br />
3c. If x is in Nil(R) and u is a unit, show that u+x is a unit.<br />
<br />
3d. Show that Nil(S), where S is the 2x2 matrices with real coefficients, is not an ideal of S.<br />
<br />
<br />
4a. If M is an R-module and I is an ideal of R such that im = 0 for all i in I and m in M, show that M carries the structure of an R/I-module.<br />
<br />
4b. If S is a commutative ring with 1 and J is a maximal ideal of S, show that J/J^2 is a vector space over S/J.<br />
<br />
<br />
5. A "divisible element" x of a Z-module is one such that for each positive integer k, there exists a y such that ky=x. Show that a free Z-module cannot contain a nonzero divisible element.<br />
<br />
<br />
6. Prove that the direct sum of any collection of free R-modules is also a free R-module.<br />
<br />
<br />
7. In this problem you will prove that the module M given by the direct product of copies of Z, indexed by the positive integers, is not a free Z-module. Therefore, assume that M is a free Z-module with basis B, and let N be the direct sum of copies of Z, as a submodule of M.<br />
<br />
7a. Show there is a countable subset B1 of B such that N is contained in the submodule N1 generated by B1. [Hint: N is countable.]<br />
<br />
7b. Show that M/N1 is a free Z-module.<br />
<br />
7c. Let S = {(b1, b2, b3, ...) : b_i = i! or -i!}. Show that there is some s in S which is not in N1. [Hint: S is uncountable.]<br />
<br />
7d. For any s in S but not N1, show that for each positive integer k, there exists some m in M with s-km in N1. Conclude that s+N1 is a divisible element of M/N1.<br />
<br />
7e. Use problem 5 to obtain a contradiction.<br />
<br />
<br />
8. In class I said "rank" is not always well-defined for R-modules if R is not commutative. Here is an example demonstrating this assertion. Let M be the countably infinite direct product of copies of Z as in exercise 7, and let R be the endomorphism ring of M. Define phi_1(a1,a2,a3,a4,...) = (a1,a3,a5,...) and phi_2(a1,a2,a3,a4,...) = (a2,a4,a6,...).<br />
<br />
8a. Show that {phi_1, phi_2} is a free basis for R, as a left R-module. [Hint: let psi_1(a1,a2,a3,...) = (a1,0,a2,0,...) and psi_2(a1,a2,a3,...) = (0,a1,0,a2,...). Compute the products of the phi_i and psi_j in either order, and use the result to show that the phi_i are a basis.]<br />
<br />
8b. Show that R is isomorphic to R^n for every positive integer n.<br />
<br />
<br />
==HOMEWORK 10 (due Dec 4)==<br />
<br />
1. '''How algebraists do calculus'''. The ''ring of dual numbers'' is the ring R = C[e]/(e^2). We think of e as an infinitesimal number; that is, something so small that its square is 0. This algebraic notion of infinitesimal is the basis for the algebraic notion of differentiation, which works as follows. Let f be a polynomial in R[x]. Show that there is a unique polynomial g in C[x] such that<br />
<br />
f(x+e) - f(x) = eg(x)<br />
<br />
and that in fact g is the derivative of f. In other words, the usual definition of "derivative" works just fine in this context, without any use of the notion of limit!<br />
<br />
2. Write down all the ideals of C[e]/e^2. (There are three.)<br />
<br />
3. Show that any right ideal of the ring M_2(Q) of 2x2 matrices is an ideal of the form I_V described in class for some subspace V of Q^2; that is, it consists precisely of those linear transformations whose image is contained in V.<br />
<br />
<br />
4. An ''idempotent'' element of a ring is an element e satisfying e^2 = e.<br />
<br />
4a. If e is an idempotent, show that 1-e is also an idempotent.<br />
<br />
4b. Suppose R has a unit (we are actually always supposing this, but I just want to remind you of it for this problem) and suppose e is a ''central'' idempotent, i.e. an idempotent which commutes with all elements of R. Show that eR and (1-e)R are both two-sided ideals. Moreover, show that eR and (1-e)R are also subrings of R (with the warning that the identity in eR is not 1, which is not contained in eR, but rather e.)<br />
<br />
4c. Show that the map f_e: R -> eR defined by f_e(r) = er is a ring homomorphism. Finally show that the ring homomorphism<br />
<br />
f_{e} x f_{1-e}: R -> eR x (1-e)R <br />
<br />
is actually an isomorphism of rings. In other words, the presence of a central idempotent induces a decomposition a ring as a direct product.<br />
<br />
<br />
5. The ''center'' of a ring R is the subring of elements commuting with every element of R.<br />
<br />
5a. Show that the center of M_n(Q) is the ring of scalar matrices.<br />
<br />
5b. Show that the center of the group ring C[G] is the set of elements sum f(g) g such that f(g) is a class function.<br />
<br />
<br />
6. '''How algebraists do differential equations.''' We define the ''Weyl algebra'' W of differential operators in one variable as follows. Multiplication by x is a linear transformation on the complex vector space V of all smooth functions on C. Differentiation, which we denote d, is also a linear transformation on V. So we let W be the subring of End(V) generated by x and d. We note that x and d satisfy the relation xd = dx - 1; in particular, the Weyl algebra is not commutative.<br />
<br />
6a. Describe the submodule of V which is annihilated by the element d-x in W.<br />
<br />
6b. Show that, for any polynomial f(x), fd-df = -f', and also show that for any polynomial g(d), gx-xg = -g'. (This is another way algebraists do calculus!) <br />
<br />
6c. Use the facts from (b) to show that the center of the Weyl algebra is just the field of constants. (Remark: you will use in the course of this proof the fact that a polynomial whose derivative is 0 is constant -- this would not be true if we were working in positive characteristic, as the example of x^p considered as a polynomial over F_p demonstrates.)<br />
<br />
<br />
7. If M is a left module for a ring R, we denote by Ann(M) the set of elements r in R such that rm = 0 for all m in M. Show that Ann(M) is a two-sided ideal. <br />
<br />
<br />
8. Let M be a left module for R, and let m be an element of M. We denote by Ann(m) the set of elements r in R such that rm = 0.<br />
<br />
8a. Show that Ann(m) is a left ideal, and show that the submodule Rm of M generated by m is isomorphic to the left module R/Ann(m).<br />
<br />
8c. Let V,W be as in question 6. Show that Ann(e^x) is the left ideal W(d-1).<br />
<br />
==HOMEWORK 11 (due Dec 13)==<br />
<br />
1. Let f: R -> S be a homomorphism of rings. As discussed in class, if M is a left R-module, then S tensor_R M is a left S-module, and if N is an S-module, then N can also be considered as an R-module by pullback of the action.<br />
<br />
Prove that if M is a finitely generated left R-module, then S tensor_R M is a finitely generated left S-module. On the other hand, give an example to show that N can be finitely generated as a left S-module but not finitely generated when considered as a left R-module.<br />
<br />
2. Recall from class that when V and W are complex vector spaces, W^dual tensor V is naturally isomorphic to the space of linear homomorphisms Hom(W,V).<br />
<br />
Now suppose that V and W are left C[G]-modules, i.e. complex representations of a group G. Show that W^dual tensor_C[G} V is the maximal G-invariant quotient of Hom(W,V); in other words, the quotient of Hom(W,V) by the subspace generated by all functions F of the form F(w) = f(gw) - gf(w) for some function f in Hom(W,V) and some g in G.<br />
<br />
3. '''How algebraists do logic.''' Why are Boolean rings called Boolean rings? Because they model some features of the Boolean arithmetic that governs propositional logic. Let A be a collection of propositions. We define two operations on A as follows. By a + b we mean the negation of a xor b, while by ab we mean "a or b."<br />
<br />
3a. It is not a priori obvious that these operations satisfy the ring axioms, but assume this for the moment. Show that a^2 = a for all a (so A is a boolean ring) and 2a = 0 for all a (so A has characteristic 2.)<br />
<br />
3b. Suppose we have a consistent assignment of truth-values to the propositions in A. Show that the set of true statements of A forms an ideal of A.<br />
<br />
3c. (Optional for people who like propositional logic) Show that a(b+c) = ab + ac with the given operations.<br />
<br />
<br />
4. Let K be Q[x]/(x^3-2).<br />
<br />
4a. Show that K is a field.<br />
<br />
4b. Show that K tensor_Q K decomposes as the direct sum of two fields. What are they? [Note: K tensor_Q K is a ring!]<br />
<br />
5. There was some discussion in class about when we can "go back and forth" between right modules and left modules, as is the case with group rings.<br />
<br />
5a. If R is a ring, we denote R^op ("the opposite ring") to be the ring whose elements are those of R, but whose multiplication law x_opp is defined by s x_opp r = rs. Show that if M is a left R-module, then the map M x R^op -> M sending (x,r) to rx makes M into a right R^op module. In particular, if there is a RING ISOMORPHISM from R to R^op, then every left R-module can be thought of as a right R-module.<br />
<br />
5b. Verify that the map sending g to g^{-1} extends by linearity to give a ring isomorphism between C[G] and C[G]^op.<br />
<br />
5c. Construct a ring isomorphism between M_n(Q) and M_n(Q)^op.<br />
<br />
5d. Optional: Give an example of a ring (with 1) which is not isomorphic to its opposite ring. (This is not so easy, I think! I found a MathOverflow thread with several examples, none of which came with a short proof of the non-isomorphism -- try your hand!)<br />
<br />
6. . Let R be a commutative ring with 1. If R has a unique maximal ideal M, R is called a "local ring".<br />
<br />
6a. If R is a local ring, show that every element of R that is not in M is a unit.<br />
<br />
6b. Conversely, if R is such that the set of nonunits forms an ideal I, show that R is a local ring with maximal ideal I.<br />
<br />
7. The inclusion of S_3 into S_4 associated to the inclusion of {1,2,3} into {1,2,3,4} induces a ring homomorphism from C[S_3} to C{S_4]. Let V be a representation of C{S_3]. Then, as discussed in class, V tensor_C[S_3] C[S_4] is a representation of S_4, called the '''induced representation''' of V.<br />
<br />
7a. Show that dim_C (V tensor_C[S_3] C[S_4]) = 4 dim_C V.<br />
<br />
7b. When V is the trivial representation of S_3, decompose (V tensor_C[S_3] C[S_4]) as a sum of irreducible representations of S_4.<br />
<br />
--><br />
<br />
<!--==HOMEWORK PROBLEMS TO GIVE IN FUTURE SEMESTERS==<br />
<br />
Let f: C[x] tensor_C C[y] -> C[z] be the map of rings with f(x) = f(y) = z. Show that f(m tensor n) is not 0 for any pure tensor m tensor n, but f is not injective. (This is a warning not to try to check injectivity by checking that no pure tensors are killed by the map.)<br />
<br />
--></div>Dummithttps://hilbert.math.wisc.edu/wiki/index.php?title=741&diff=62537412013-11-09T01:30:43Z<p>Dummit: /* HOMEWORK 9 (Due Nov 13) */</p>
<hr />
<div>'''Math 741'''<br />
<br />
Algebra<br />
<br />
Prof: [http://www.math.wisc.edu/~ellenber Jordan Ellenberg]<br />
<br />
Grader: Evan Dummit.<br />
* Homework policies:<br />
* Late homework may be given directly to the grader, along with either <br />
(i) the instructor's permission, or (ii) a polite request for mercy.<br />
* Assignments that are more than 1 page should be affixed in some reasonable way.<br />
* Results from places (e.g., the internet) other than 741 and standard books must be cited.<br />
<br />
JE's office hours: Monday 12pm-1pm (right after class)<br />
<br />
This course, the first semester of the introductory graduate sequence in algebra, will cover the basic theory of groups, group actions, representation, linear and multilinear algebra, and the beginnings of ring theory.<br />
<br />
==SYLLABUS==<br />
<br />
In this space we will record the theorems and definitions we covered each week, which we can use as a list of notions you should be prepared to answer questions about on the Algebra qualifying exam. The material covered on the homework is also an excellent guide to the scope of the course. <br />
<br />
'''WEEK 1''': <br />
<br />
Definition of group. Associativity. Inverse. <br />
<br />
Examples of group: GL_n(R). GL_n(Z). Z/nZ. R. Z. R^*. The free group F_k on k generators. <br />
<br />
Homomorphisms. The homomorphisms from F_k to G are in bijection with G^k. Isomorphisms.<br />
<br />
'''WEEK 2''':<br />
<br />
The symmetric group (or permutation group) S_n on n letters. Cycle decomposition of a permutation. Order of a permutation. Thm: every element of a finite group has finite order. <br />
<br />
Subgroups. Left and right cosets. Lagrange's Theorem. Cyclic groups. The order of an element of a finite group is a divisor of the order of the group.<br />
<br />
The sign homomorphism S_n -> +-1.<br />
<br />
<br />
'''WEEK 3'''<br />
<br />
Normal subgroups. The quotient of a group by a normal subgroup. The first isomorphism theorem. Examples of S_n -> +-1 and S_4 -> S_3 with kernel V_4, the Klein 4-group.<br />
<br />
Centralizers and centers. Abelian groups. The center of SL_n(R) is either 1 or +-1.<br />
<br />
Groups with presentations. The infinite dihedral group <x,y | x^2 = 1, y^2 = 1>.<br />
<br />
'''WEEK 4'''<br />
<br />
More on groups with presentations.<br />
<br />
Second and third isomorphism theorems.<br />
<br />
Semidirect products.<br />
<br />
'''WEEK 5'''<br />
<br />
Group actions, orbits, and stabilizers.<br />
<br />
Orbit-stabilizer theorem.<br />
<br />
Cayley's theorem.<br />
<br />
Cauchy's theorem.<br />
<br />
'''WEEK 6'''<br />
<br />
Applications of orbit-stabilizer theorem (p-groups have nontrivial center, first Sylow theorem.)<br />
<br />
Classification of finite abelian groups and finitely generated abelian groups.<br />
<br />
Composition series and the Jordan-Holder theorem (which we state but don't prove.) <br />
<br />
The difference between knowing the composition factors and knowing the group (e.g. all p-groups of the same order have the same composition factors.)<br />
<br />
'''WEEK 7'''<br />
<br />
Simplicity of A_n.<br />
<br />
Nilpotent groups (main example: the Heisenberg group)<br />
<br />
Derived series and lower central series.<br />
<br />
Category theory 101: Definition of category and functor. Some examples. A group is a groupoid with one object.<br />
<br />
'''WEEK 8'''<br />
<br />
Introduction to representation theory.<br />
<br />
'''WEEK 10'''<br />
<br />
Ring theory 101: Rings, ring homomorphisms, ideals, isomorphism theorems. Examples: fields, Z, the Hamilton quaternions, matrix rings, rings of polynomials and formal power series, quadratic integer rings, group rings. Integral domains. Maximal and prime ideals. The nilradical.<br />
<br />
Module theory 101: Modules, module homomorphisms, submodules, isomorphism theorems. Noetherian modules and Zorn's lemma. Direct sums and direct products of arbitrary collections of modules.<br />
<br />
<br />
==HOMEWORK 1 (due Sep 16)==<br />
<br />
1. Suppose that H_1 and H_2 are subgroups of a group G. Prove that the intersection of H_1 and H_2 is a subgroup of G.<br />
<br />
2. Recall that S_3 (the symmetric group) is the group of permutations of the set {1..3}. List all the subgroups of S_3.<br />
<br />
3. We can define an equivalence relation on rational numbers by declaring two rational numbers to be equal whenever they differ by an integer. We denote the set of equivalence classes by Q/Z. The operation of addition makes Q/Z into a group.<br />
<br />
a) For each n, prove that Q/Z has a subgroup of order n.<br />
<br />
b) Prove that Q/Z is a ''divisible'' group: that is, if x is an element of Q/Z and n is an integer, there exists an element y of Q/Z such that ny = x. (Note that we write the operation in this group as addition rather than multiplication, which is why we write ny for the n-fold product of y with itself rather than y^n)<br />
<br />
c) Prove that Q/Z is not finitely generated. (Hint: prove that if x_1, .. x_d is a finite subset of Q/Z, the subgroup of Q/Z generated by x_1, ... x_d is finite.)<br />
<br />
d) Conclude that Q is not finitely generated.<br />
<br />
4. We will prove that there is no homomorphism from SL_2(Z) to Z except the one which sends all of SL_2(Z) to 0. Suppose f is a homomorphism from SL_2(Z) to Z.<br />
<br />
a) Let U1 be the upper triangular matrix with 1's on the diagonal and a 1 in the upper right hand corner, as in class, and let U2 be the transpose of U1, also as in class. Show that (U1 U2^{-1})^6 = identity (JING 1, TAO 0) and explain why this implies that f(U1) = f(U2).<br />
<br />
b) Show that there is a matrix A in SL_2(Z) such that A U1 A^{-1} = U2^{-1}. (Recall that we say U1 and U2^{-1} are "conjugate".) Explain why this also implies that f(U1) = -f(U2).<br />
<br />
c) Explain why a) and b) imply that f must be identically 0.<br />
<br />
5. The argument above also shows that there is no nonzero homomorphism from SL_2(Z) to Z/pZ where p is a prime greater than 3. However, it leaves open the possibility that there is indeed a nonzero homomorphism from SL_2(Z) to Z/2Z. Exhibit such a homomorphism. Optional challenge problem: exhibit a nonzero homomorphism from SL_2(Z) to Z/3Z.<br />
<br />
<br />
==HOMEWORK 2 (due Sep 25)==<br />
<br />
1. Let V be the Klein 4-group in S_4. Let Q be the symmetric group on the set DF of double-flips in S_4; there are 3 double flips, so Q is isomorphic to S_3. If g is an element of S_4, we discussed in class that conjugation by g permutes the elements of DF. So to each element of g, we have associated an element f(g) of Q. More or less by definition, this defines a homomorphism f: G -> Q. Show that this homomorphism is surjective and has kernel H, and thus that G/H is isomorphic to Q.<br />
<br />
2. Let F_2 be the free group on two generators, which we denote x,y. Prove that there is a automorphism of F_2 which sends x to xyx and y to xy, and prove that this automorphism is unique.<br />
<br />
3. Let H be a subgroup of G, and let N_G(H), the ''normalizer'' of H in G, be the set of elements g in G satisfying g H g^{-1} = H. Prove that N_G(H) is a subgroup of G.<br />
<br />
4. Let T be the subgroup of GL_2(R) consisting of diagonal matrices. This is an example of a "Cartan subgroup," or "torus" (whence the notation T.) Describe the normalizer N(T) of T in GL_2(R). Prove that there is a homomorphism from N(T) to Z/2Z whose kernel is precisely T. <br />
<br />
5. Prove that the group of inner automorphisms of a group is normal in the full automorphism group Aut(G).<br />
<br />
6. Let H and H' be subgroups of a group G, and let x be an element of G. We denote by HxH' the set of elements of the form hxh', with h in H and h' in H', and we call HxH a ''double coset'' of the pair (H,H').<br />
<br />
a) Show that G decomposes as a disjoint union of double cosets of (H,H').<br />
<br />
b) Let G be GL_2(R) and let B be the subgroup of upper triangular matrices. Prove that G decomposes into exactly two double cosets of (B,B). (This is an example of the so-called *Bruhat decomposition* which is of great importance in the theory of algebraic groups and their representations.)<br />
<br />
c) Let G be the symmetric group S_n, and let H be the subgroup of permutations which fix 1. Describe the double coset decomposition of S_n into double cosets of (H,H).<br />
<br />
<br />
==HOMEWORK 3 (due Oct 2)==<br />
<br />
1. Compute the centralizer of the triple flip (12)(34)(56) in S_6. (Note: this is the kind of problem that a computer algebra package like sage is very good at, and I encourage you to learn to use sage well enough to check your answer.)<br />
<br />
2. Warning: a normal subgroup of a normal subgroup need not be normal! Let H be the subgroup of S_4 generated by (12)(34). Then H is a subgroup of the Klein 4-group V_4, which we have already shown is normal in S_4. Show that H is normal in V_4, but H is not normal in S_4.<br />
<br />
3. Let G be a group and let H_1 and H_2 be subgroups of G of finite index (that is, the number of cosets in G/H_i is finite, i = 1,2.) Prove that H_1 intersect H_2 also has finite index in G.<br />
<br />
4. Let X be the set of lines in F_3^2 (here a line means a 1-dimensional linear subspace) so that |X| = 4. Let G be the projective linear group PGL_2(F_3), so that G acts on X via its action on F_3^2. This action can be thought of as a homomorphism from G to S_4. Show that this homomorphism is an ISOMORPHISM from PGL_2(F_3) to S_4.<br />
<br />
5. We proved two theorems in class: Lagrange's theorem, which says that the order of a subgroup of G divides |G|, and Cauchy's theorem, which says that if p divides |G| then there exists a subgroup of G of order p. Cauchy's theorem is a kind of partial converse to Lagrange's theorem, but the full converse is false. Give an example of a finite group G and a divisor n of |G| such that you can prove there is no subgroup of G of order n.<br />
<br />
6. The group S_3 has only three conjugacy classes. Prove that a finite group with at most three conjugacy classes has order at most 6.<br />
<br />
7. Let X be the set of ordered triples of elements of {1,..,n}, for some n >= 3. Then S_n acts on X. How many orbits are there? (Hint: the answer does not depend on n.) OPTIONAL: How many orbits are there on the space of ordered k-tuples, when n >= k? <br />
<br />
==HOMEWORK 4 (due Oct 9)==<br />
<br />
1. Recall that the commutator [x,y] is x y x^{-1} y^{-1}. If G is a group, the commutator subgroup of G, denoted G' or [G,G], is the subgroup of G generated by all commutators [x,y] for x,y in G.<br />
<br />
1a. Show that G' is a normal subgroup of G.<br />
1b. Show that G/G' is an abelian group, which we call the abelianization, G^ab.<br />
1c. Show that if f: G -> A is a homomorphism, with A abelian, then G' is contained in ker f. Conclude that f factors through a homomorphism G/G' -> A.<br />
1d. Show that G' = G if and only if G has no nontrivial abelian quotient. In this case, we say G is ''perfect''.<br />
<br />
2. Prove that any p-Sylow subgroup of GL_n(F_p) is contained in SL_n(F_p).<br />
<br />
3. Let U_n(F_p) be the subgroup of GL_n(F_p) consisting of upper triangular matrices with 1's on the diagonal. Show that U_n(F_p) is a p-Sylow subgroup of GL_n(F_p).<br />
<br />
4. U_n(F_p) is called the *unipotent subgroup* of GL_n(F_p), because it has the property (not part of your homework, but easy to check) that every u in U_n(F_p) is unipotent, i.e. its characteristic polynomial is (x-1)^n. It is a p-group (as you know by virtue of the last problem) so it must have a nontrivial center. What is the center of U_nF_p)?<br />
<br />
5. U_3(F_p) is Shamgar Gurevich's favorite group, the *Heisenberg group" of order p^3. Write down a composition series for U_3(F_p).<br />
<br />
6. Let A be the subgroup of Z^2 generated by the vector (1,0). Let B be the subgroup of Z^2 generated by the vector (3,3). Show that A and B are isomorphic to each other, but Z^2 / A is not isomorphic to Z^2 / B.<br />
<br />
7. Suppose that G and H are groups and f: G -> H is a homomorphism. Recall the definition of the abelianization G^ab from problem 1.<br />
<br />
7a. Show that the composition G -> H -> H^ab factors through a unique homomorphism G^ab -> H^ab, which we denote f^ab.<br />
7b. Show that if f: G -> H and g: H -> Q are homomorphisms, then f^ab o g^ab = (f o g)^ab.<br />
<br />
(For those reading MacLane, this constitutes a proof that abelianization is a '''functor''' from the category of groups to the category of abelian groups.)<br />
<br />
8. Give two different composition series for S_4 and show that they have the same composition factors.<br />
<br />
OPTIONAL: We saw in class that a finitely generated abelian group that was torsion-free was in fact a free abelian group. This totally dies without the hypothesis of abelianness. To see this, give an example of a finitely generated group which is torsion free but which is not a free group, and which does not even *contain* any free group of rank greater than 1.<br />
<br />
==HOMEWORK 5 (due Oct 16)==<br />
<br />
1. Let G be a group, thought of as a category with one object. Show that a set with an action of G is the same thing as a functor from G to the category of sets.<br />
<br />
2. If (V, rho) is a representation of a group G, denote by V^G the space of vectors in V such that gv = v for all g in G; these are called _invariant_ vectors. We saw that the permutation representation of S_3 has a one-dimensional space of invariants. Prove that if X is a finite set with G-action, and V_X the corresponding permutation representation of G, then<br />
<br />
dim V_X^G = number of orbits of X.<br />
<br />
3. (Invariants are functorial) Let G be a group. Then there is a category called C[G]-Mod whose objects are complex representations of G (i.e. pairs (V,rho) where V is a complex vector space and rho is a homomorphism from G to GL(V)). The morphisms in this category are the ones described in class: Mor((V,rho),(W,psi)) is the set of linear maps f from V to W such that<br />
<br />
f(rho(g)(v)) = psi(g)(f(v))<br />
<br />
for all g in G and all v in V.<br />
<br />
Show that there is a functor H_0 from the category of representations of G to the category of vector spaces whose action on objects is given by<br />
<br />
H_0((V) = V^G.<br />
<br />
(In other words: your job again is to explain how to associate to a morphism from (V,rho) to (W,phi) a morphism from V^G to W^G, in a way that's compatible with composition.)<br />
<br />
4. Let G be a finite group, let X be a set with G-action, and let chi be the character of the corresponding permutation representation. Show that, for each g in G, chi(g) is the number of elements of X fixed by g. (I am actually going to prove this in class.)<br />
<br />
5. Let V be the space of homogeneous degree-3 polynomials in variables x_1, x_2, x_3; then S_3 acts on V by permutation of the three variables. Describe the decomposition of V into irreducible representations of S_3.<br />
<br />
6a. Show that if chi is the character of a representation of a finite group G, then chi(g^{-1}) is the complex conjugate of chi.<br />
<br />
6b. As a consequence, show that if g is conjugate to g^{-1} for all g in G, then chi(g) is real.<br />
<br />
6c. The groups D_p and S_n have the property that g is conjugate to g^{-1} for _every_ g in G, so that every character is a real-valued function on G. Give an example of a group G where this is not the case, and give an example of a representation of G whose character takes non-real values.<br />
<br />
OPTIONAL: (for people who know some topology.) Let $X$ be a topological space and let $Paths/X$ be the category of paths up to isotopy as described in class. Explain how to construct a bijection between {functors from Paths/X to the category FinSet of finite sets} and {finite covering spaces of X}. (For simplicity you may assume X is path-connected.)<br />
<br />
==HOMEWORK 6 (due Oct 23)==<br />
<br />
1. Let V_k be the space of homogenous degree-k polynomials in x_1, x_2, x_3. Compute the character of S_3 acting on V_k, and, using this, give formulas for the multiplicity of the trivial rep, the sign rep, and the standard rep of S_3 in V_k.<br />
<br />
2. (Schur's lemma can be false over non-algebraically closed fields) Let G be Z/3Z and let g be a generator of G. Then G acts by cyclic permutations on the set {1,2,3}; let P be the corresponding permutation representation on the REAL (not complex!) vector space generated by e_1,e_2,e_3.<br />
<br />
2a. Show that P breaks up as the direct sum of the trivial representation and a 2-dimensional representation V which is irreducible. (Remember, we are not working over the complex numbers, so you cannot use the criterion for irreducibility in terms of the character that we worked out in class for the complex case; you need to prove directly that V has no nontrivial proper subrepresentation.)<br />
<br />
2b. Show that the map from V to V induced by g is G-equivariant, but is neither zero nor a scalar multiplication.<br />
<br />
<br />
The following two problems involve the tensor product. I am going to teach the general notion of tensor product of modules in this course, but I am expecting you already know what the tensor product of vector spaces. If not, please review it by learning the definition and doing these problems, because having a familiarity with the vector space theory will be very helpful in learning the general story.<br />
<br />
3. Let V and W be complex vector spaces. Let f be a linear transformation of V, and g a linear transformation of W. <br />
<br />
3a. Show that there is a unique linear transformation F satisfying<br />
<br />
F(v tensor w) = f(v) tensor g(w)<br />
<br />
for all v in V and all w in W. We denote this transformation by f tensor g.<br />
<br />
3b. Suppose V and W are finite dimensional. Show that Trace(f tensor g) = Trace(f) Trace(g). What does this say when f and g are both the identity transformation?<br />
<br />
4. Let V be a vector space. We define Sym^2 V to be the quotient of V tensor V by the subspace generated by all elements of the form<br />
<br />
v tensor w - w tensor v<br />
<br />
for v,w in V.<br />
<br />
Suppose dim V = n. What is dim Sym^2 V?<br />
<br />
5. For any n, the group S_n has an (n-1)-dimensional representation, the "standard representation," on the space of vectors in C^n whose coordinates sum to 0. When n=3, we proved in class that this representation was irreducible. Prove that the standard representation is irreducible for all n.<br />
<br />
6. Give an example showing that the standard representation of S_n need NOT be irreducible over a field of characteristic p.<br />
<br />
7. Let H be a subgroup of G and let V be a representation of G. Let chi_{G/H} be the character of the permutation representation of G on the set of cosets G/H, so that, by problem 1, chi_{G/H}(g) is the number of fixed points of g in its action on the cosets. As discussed in class, the dimension of the G-invariant subspace V^G is given by the inner product of chi_V with the trivial character. Prove that the dimension of V^H is given by the inner product of chi_V with chi_{G/H}.<br />
<br />
==HOMEWORK 7 (due Oct 30)==<br />
<br />
1. When a formula always yields a non-negative integer, it is often because it is secretly a formula for either the cardinality of some finite set or the dimension of some finite-dimensional vector space. In this case, show that<br />
<br />
<chi_{V_1}, chi_{V_2}> = dim_C Hom_G(V_1,V_2)<br />
<br />
where we recall that Hom_G(V_1,G_2) refers to the space of G-equivariant homomorphisms from V_1 to V_2.<br />
<br />
2. Let G be the group of affine linear transformations of F_5; that is, it is the group of transformations x -> ax + b where a lies in (Z/5Z)^* and b lies in (Z/5Z). Note that G has order 20. Write H for the subgroup of G consisting of transformations fixing 0 (i.e. those for which b=0). <br />
<br />
We will work out all the irreducible representations of G. <br />
<br />
2a. First of all, consider the 5-dimensional permutation representation of G acting by left multiplication on the cosets of H. Show that this representation decomposes as a direct sum of the trivial representation and an irreducible 4-dimensional representation. (Hint: it is probably easier to use the character of this representation than to prove irreducibility directly.) <br />
<br />
2b. Now show that there are 4 1-dimensional representations of G.<br />
<br />
2c. Show that the irreducible representations of G you have constructed are the only ones.<br />
<br />
3. If G and H are finite groups, and V is a representation of G, and W is a representation of H, then V tensor W is a representation of GxH, and the character chi_{V tensor W} is given by<br />
<br />
chi_{V tensor W}(g,h) = chi_V(g) chi_W(h).<br />
<br />
3a. Prove that V tensor W is irreducible if and only if V and W are irreducible.<br />
<br />
3b. Prove that all the irreducible representations of G x H are of the form V tensor W.<br />
<br />
(Together, these propositions show that you can completely describe the representation theory of G x H in terms of that of G and that of H.)<br />
<br />
4. Suppose that V is an irreducible representation of S_n, and suppose that when we consider V as a representation of the alternating group A_n it is NOT irreducible. Prove that V, considered as a representation of A_n, is the direct sum of two non-isomorphic irreducible representations. Prove furthermore that chi_V(g) = 0 for all odd permutations g. Give an example of such an irreducible representation of S_4.<br />
<br />
<br />
OPTIONAL (because I couldn't quickly think of an easy way to do it!) Prove that there exists such a representation of S_n for every n >= 3.<br />
<br />
5. (Fourier analysis over finite fields.) I said in class that the representation theory that goes into Fourier analysis is different from the representation theory of finite groups we do in class, but that's not quite true; when you do Fourier theory over finite fields, the two theories come into much closer contact, with the bonus that we don't have to worry about issues of infinite sums that I hand-waved away in class.<br />
<br />
Let F be the field of p elements and let V be the space of complex-valued functions on F; so V is a p-dimensional space. Let G be the group Z/pZ (in other words, it is the additive group of the field.)<br />
<br />
Now V is a representation of G, in the same way we discussed in class: if f is a function in V, and a is an element of g, then the function gf is defined by<br />
<br />
gf(x) = f(x+a)<br />
<br />
5a. Describe the breakup of V into irreducible representations of G, which are all 1-dimensional (not just what the characters are, explicitly decribe the irreducible constituents of V as subspaces of V!)<br />
<br />
5b. There is a natural norm on V which sends f to ||f|| = sum_x |f(x)|^2.<br />
<br />
Given any f, we have a unique decomposition f = sum_i f_i, where f_i lies in the irreducible constituent V_i of V.<br />
<br />
Prove that ||f|| = sum_i ||f_i||.<br />
<br />
OPTIONAL (for people who are taking analysis) Explain why ||f_i|| is correctly thought of as a "Fourier coefficient" and why the fact proved in 5b is the finite-field analogue of Parseval's identity.<br />
<br />
==HOMEWORK 8 (due Nov 6)==<br />
<br />
1. If R is a ring (not necessarily with 1) such that a^2 = a for every a in R, R is called a Boolean ring.<br />
<br />
1a. Show that any nonzero Boolean ring has characteristic 2. (In other words, show that a+a=0 for all a in R.)<br />
<br />
1b. Show that every Boolean ring is commutative.<br />
<br />
2. Let phi : R -> S be a homomorphism of commutative rings (recalling that rings for us have a multiplicative identity 1 and that ring homomorphisms take 1 to 1.)<br />
<br />
2a. If I is an ideal of S, show that phi^(-1) (I) is an ideal of R.<br />
<br />
2b. If P is a prime ideal of S, show that phi^(-1) (P) is either R or a prime ideal of R.<br />
<br />
2c. If M is a maximal ideal of S and phi is surjective, show that phi^(-1) (M) is a maximal ideal of R.<br />
<br />
2d. Find a homomorphism phi : R -> S and a maximal ideal M of S such that phi^(-1) (M) is not a maximal ideal of R.<br />
<br />
3. Let Nil(R) denote the set of nilpotent elements of a ring R; that is, the set of x such that x^k = 0 for some positive integer k. <br />
<br />
3a. Show that Nil(R) is an ideal whenever R is a commutative ring. Assume R is commutative for the remainder of this exercise.<br />
<br />
3b. Describe Nil(Z/720Z).<br />
<br />
3c. If x is in Nil(R) and u is a unit, show that u+x is a unit.<br />
<br />
3d. Show that Nil(M_2(R)) is not an ideal of the matrix algebra M_2(R). (That is, it is neither a left nor a right ideal.)<br />
<br />
4. We say a ring is Noetherian if any ascending chain of ideals I_1 < I_2 < I_3 < …. eventually stabilizes. This is a somewhat funny-looking condition if you haven't encountered it before, but it turns out to be a very useful way of formalizing the notion that a ring is "not too bad."<br />
<br />
4a. Show that Z is Noetherian.<br />
<br />
4b. Let R be the ring of continuous functions from the real numbers to the real numbers. Show that R is not Noetherian.<br />
<br />
5. '''How algebraists do calculus'''. The ''ring of dual numbers'' is the ring A = C[e]/(e^2). We think of e as an infinitesimal number; that is, something so small that its square is 0. This algebraic notion of infinitesimal is the basis for the algebraic notion of differentiation, which works as follows. Let f be a polynomial in C[x]. Show that there is a unique polynomial g in C[x] such that<br />
<br />
f(x+e) - f(x) = eg(x)<br />
<br />
and that in fact g is the derivative of f. In other words, the usual definition of "derivative" works just fine in this context, without any use of the notion of limit!<br />
<br />
6. Write down all the ideals of C[e]/e^2. (There are three.)<br />
<br />
7. Show that any right ideal of the ring M_2(Q) of 2x2 matrices is an ideal of the form I_V described in class for some subspace V of Q^2; that is, it consists precisely of those linear transformations whose image is contained in V.<br />
<br />
== HOMEWORK 9 (Due Nov 13) ==<br />
<br />
1. If M is an R-module and I is an ideal of R such that im = 0 for all i in I and m in M, show that M carries the structure of an R/I-module.<br />
<br />
2. It follows from 1. that if S is a commutative ring and J is a maximal ideal of S, that J/J^2 is a vector space over the field S/J. Compute the dimension of J/J^2 as a vector space over S/J when<br />
<br />
2a. S = Z, J = (5);<br />
2b. S = C[x,y], J = (x,y)<br />
<br />
2c. Suppose S = C[x,y]/(f) and J = (x,y). Then the dimension of J/J^2 depends on the choice of the polynomial f. Compute the possible values of this dimension and give an example of an f realizing each possibility. (NOTE: this is secretly, or not-so-secretly, another instance of differential geometry as carried out by algebraists...)<br />
<br />
3. An ''idempotent'' element of a ring is an element e satisfying e^2 = e.<br />
<br />
3a. If e is an idempotent, show that 1-e is also an idempotent.<br />
<br />
3b. Suppose R has a unit (we are actually always supposing this, but I just want to remind you of it for this problem) and suppose e is a ''central'' idempotent, i.e. an idempotent which commutes with all elements of R. Show that eR and (1-e)R are both two-sided ideals. Moreover, show that eR and (1-e)R are also subrings of R (with the warning that the identity in eR is not 1, which is not contained in eR, but rather e.)<br />
<br />
3c. Show that the map f_e: R -> eR defined by f_e(r) = er is a ring homomorphism. Finally show that the ring homomorphism<br />
<br />
f_{e} x f_{1-e}: R -> eR x (1-e)R <br />
<br />
is actually an isomorphism of rings. In other words, the presence of a central idempotent induces a decomposition a ring as a direct product.<br />
<br />
4. Let k be an algebraically closed field, and let D be a finite-dimensional division algebra over k. Prove that D = k. (Hint: if alpha is an element of D which is not in k, prove that there is some polynomial f(x) such that f(alpha) = 0. Then think back to our proof in class that the quaternion algebra with complex coefficients was not a division algebra.)<br />
<br />
5. The ''center'' of a ring R is the subring of elements commuting with every element of R.<br />
<br />
5a. Show that the center of M_n(Q) is the ring of scalar matrices.<br />
<br />
5b. Show that the center of the group ring C[G] is the set of elements sum f(g) g such that f(g) is a class function. Compute the central idempotents of C[S_3].<br />
<br />
6. '''How algebraists do differential equations.''' We define the ''Weyl algebra'' W of differential operators in one variable as follows. Multiplication by x is a linear transformation on the complex vector space V of all smooth functions on C. Differentiation, which we denote d, is also a linear transformation on V. So we let W be the subring of End(V) generated by x and d. We note that x and d satisfy the relation xd = dx - 1; in particular, the Weyl algebra is not commutative.<br />
<br />
6a. Describe the submodule of V which is annihilated by the element d-x in W.<br />
<br />
6b. Show that, for any polynomial f(x), fd-df = -f', and also show that for any polynomial g(d), gx-xg = -g'. (This is another way algebraists do calculus!) <br />
<br />
6c. Use the facts from 6b. to show that the center of the Weyl algebra is just the field of constants. (Remark: you will use in the course of this proof the fact that a polynomial whose derivative is 0 is constant -- this would not be true if we were working in positive characteristic, as the example of x^p considered as a polynomial over F_p demonstrates.)<br />
<br />
7. If M is a left module for a ring R, we denote by Ann(M) the set of elements r in R such that rm = 0 for all m in M. Show that Ann(M) is a two-sided ideal. <br />
<br />
8. Let M be a left module for R, and let m be an element of M. We denote by Ann(m) the set of elements r in R such that rm = 0.<br />
<br />
8a. Show that Ann(m) is a left ideal, and show that the submodule Rm of M generated by m is isomorphic to the left module R/Ann(m).<br />
<br />
8c. Let V,W be as in question 6. Show that Ann(e^x) is the left ideal W(d-1).<br />
<br />
<!--Below you will find a repository of homework problems. Note that some of these problems are taken from Lang's ''Algebra''.<br />
<br />
==HOMEWORK 1 (due Sep 20)==<br />
<br />
1. Suppose that H_1 and H_2 are subgroups of a group G. Prove that the intersection of H_1 and H_2 is a subgroup of G.<br />
<br />
2. Recall that S_3 (the symmetric group) is the group of permutations of the set {1..3}. List all the subgroups of S_3.<br />
<br />
3. We can define an equivalence relation on rational numbers by declaring two rational numbers to be equal whenever they differ by an integer. We denote the set of equivalence classes by Q/Z. The operation of addition makes Q/Z into a group.<br />
<br />
a) For each n, prove that Q/Z has a subgroup of order n.<br />
<br />
b) Prove that Q/Z is a ''divisible'' group: that is, if x is an element of Q/Z and n is an integer, there exists an element y of Q/Z such that ny = x. (Note that we write the operation in this group as addition rather than multiplication, which is why we write ny for the n-fold product of y with itself rather than y^n)<br />
<br />
c) Prove that Q/Z is not finitely generated. (Hint: prove that if x_1, .. x_d is a finite subset of Q/Z, the subgroup of Q/Z generated by x_1, ... x_d is finite.)<br />
<br />
4. Let F_2 be the free group on two generators x,y. Prove that there is a automorphism of F_2 which sends x to xyx and y to xy, and prove that this automorphism is unique.<br />
<br />
5. Let H be a subgroup of G, and let N_G(H), the ''normalizer'' of H in G, be the set of elements g in G satsifying g H g^{-1} = H. Prove that N_G(H) is a subgroup of G.<br />
<br />
6. Let T be the subgroup of GL_2(R) consisting of diagonal matrices. This is an example of a "Cartan subgroup," or "torus" (whence the notation T.) Describe the normalizer N(T) of T in GL_2(R). Prove that there is a homomorphism from N(T) to Z/2Z whose kernel is precisely T. <br />
<br />
7. Prove that the group of inner automorphisms of a group is normal in the full automorphism group Aut(G).<br />
<br />
8. Let H and H' be subgroups of a group G, and let x be an element of G. We denote by HxH' the set of elements of the form hxh', with h in H and h' in H', and we call HxH a ''double coset'' of the pair (H,H').<br />
<br />
a) Show that G decomposes as a disjoint union of double cosets of (H,H').<br />
<br />
b) Let G be GL_2(R) and let B be the subgroup of upper triangular matrices. Prove that G decomposes into exactly two double cosets of (B,B).<br />
<br />
c) Let G be the symmetric group S_n, and let H be the subgroup of permutations which fix 1. Describe the double coset decomposition of S_n into double cosets of (H,H).<br />
<br />
==HOMEWORK 2 (due Sep 27)==<br />
<br />
1. Prove that the symmetric group S_n is generated by transpositions. Hint: proceed by induction on n. Let H be the subgroup of S_n consisting of permutations fixing 1. Observe that H is isomorphic to S_{n-1}, so that H is generated by transpositions. Explain why it suffices to show that, for any g in S_n, the coset gH contains a transposition. Then explain why this is the case.<br />
<br />
2. Given that the symmetric group S_n is generated by transpositions, show that there are no nontrivial homomorphisms from S_n to Z/pZ for any odd prime p.<br />
<br />
3. Compute the centralizer of the triple flip (12)(34)(56) in S_6. (Note: this is the kind of problem that a computer algebra package like sage is very good at, and I encourage you to learn to use sage well enough to check your answer.)<br />
<br />
4. Warning: a normal subgroup of a normal subgroup need not be normal! Let H be the subgroup of S_4 generated by (12)(34). Then H is a subgroup of the Klein 4-group V_4, which we have already shown is normal in S_4. Show that H is normal in V_4, but H is not normal in S_4.<br />
<br />
5. The '''quaternion group''' Q is a non-abelian finite group of order 8. Its elements are 1, -1, i, -i, j, -j, k, -k; it satisfies g^2 = -1 for all g except +-1, and ijk = 1. (Note that the notation "-i" means "the product of i with -1" and -1 is understood to be central in Q.) Prove that every subgroup of Q is normal.<br />
<br />
6. (finishing example done in class) Let Gamma be the group F<x,y> / (x^2 = 1, y^2 = 1). (Remember, this means the quotient of the free group by the group generated by all conjugates of x^2 and y^2.) Prove that Gamma is infinite. Prove that Gamma has a subgroup R generated by xy isomorphic to Z, that R is normal, and that the quotient Gamma/R is isomorphic to Z/2Z. <br />
<br />
7. Suppose a group G has the property that every element in g satisfies g^2 = 1. Show that G must be abelian.<br />
<br />
==HOMEWORK 3 (due Oct 4)==<br />
<br />
1. Multiplication by -1 is an automorphism of Z/nZ. Let a be the homomorphism Z/2Z -> Aut(Z/nZ) which sends the nontrivial element of Z/2Z to multiplication by -1. Then we have a semidirect product of Z/pZ by Z/2Z as defined in class. This group of order 2n is called a ''dihedral group.'' and is denoted D_n (or sometimes D_{2n}).<br />
<br />
1a. Compute the center of D_n. (Note that the answer depends on n!)<br />
1b. Show that all involutions of D_n are conjugate to each other if and only if n is odd.<br />
<br />
2. (More on direct products that we didn't do in class.) Suppose that G is a semidirect product of N with H. Suppose furthermore that H is also normal in G. Prove that G is a direct product of N with H.<br />
<br />
3. Let Q be the quaternion group from last week's homework. Show that Q does not decompose as a semidirect product N \rtimes H, apart form the trivial cases N = {e} and N = Q.<br />
<br />
4. The ''affine linear group'' of degree n is the group of transformations from R^n to R^n of the form x -> Ax + b, where A is a matrix in GL_n(R) and b is a vector in R^n. Prove that the affine linear group is a semidirect product R^n \rtimes GL_n.<br />
<br />
5. The ''ordinary triangle group'' T(p,q,r) is the group with presentation <x,y | x^p = y^q = (xy)^r = 1>. This is a very interesting family of groups which arises naturally in many contexts in number theory and topology.<br />
<br />
6a. Prove that T(2,2,n) is isomorphic to the dihedral group D_n.<br />
6b. Prove that T(2,3,3) is finite and compute its order. Is it isomorphic to a finite group we've discussed in class?<br />
<br />
(It is a beautiful fact, but outside the scope of this course, that T(p,q,r) is finite if and only if 1/p + 1/q + 1/r > 1.)<br />
<br />
7. Let G be a subgroup of GL_n(R) which contains SL_n(R). Prove that G is a normal subgroup of GL_n(R).<br />
<br />
8. Let G be a group and let H_1 and H_2 be subgroups of G of finite index (that is, the number of cosets in G/H_i is finite, i = 1,2.) Prove that H_1 intersect H_2 also has finite index in G.<br />
<br />
9. Recall that the commutator [x,y] is x y x^{-1} y^{-1}. If G is a group, the commutator subgroup of G, denoted G' or [G,G], is the subgroup of G generated by all commutators [x,y] for x,y in G.<br />
<br />
9a. Show that G' is a normal subgroup of G.<br />
9b. Show that G/G' is an abelian group.<br />
9c. Show that if f: G -> A is a homomorphism, with A abelian, then G' is contained in ker f. Conclude that f factors through a homomorphism G/G' -> A.<br />
9d. Show that G' = G if and only if G has no nontrivial abelian quotient. In this case, we say G is ''perfect''.<br />
<br />
10. Under what conditions on (p,q,r) is the triangle group T(p,q,r) perfect?<br />
<br />
==HOMEWORK 4 (due Oct 16)==<br />
<br />
1. Let X be a set with a transitive action of a group G, which we think of as a homomorphism f: G -> Sym(X). Let H_x be the stabilizer of an element x of X.<br />
<br />
1a. If x' is another element of X, show that H_{x"} and H_x are conjugate subgroups. (Hint: this is not true without the hypothesis of transitivity, so make sure to use it!)<br />
<br />
1b. Show that the kernel of f is the intersection of all the conjugates of H_x in G. (This subgroup is called the ''normal core'' of H_x.)<br />
<br />
2. This week, a certified mathematical grown-up asked on MathOverflow what the index-3 subgroups of SL_2(Z/pZ) were. Let's do it ourselves for homework. Let p be a prime, and suppose H is an index-3 subgroup of SL_2(Z/pZ). I do not claim that the proof sketched below is necessarily the easiest, but it allows us to cover some ground that we won't get to do in lecture.<br />
<br />
2a. Using the coset action, show that the normal core of H is the kernel of a homomorphism f from SL_2(Z/pZ) to S_3.<br />
<br />
2b. A ''unipotent matrix'' is a matrix A such that (A-I)^n = 0 for some positive integer n. Prove that a unipotent element of SL_2(Z/pZ) has order dividing p.<br />
<br />
2c. You may use the fact that unipotent matrices generate SL_2(Z/pZ). Given this fact, show that f must be trivial when p > 3.<br />
<br />
2d. Using the above, show that SL_2(Z/pZ) has no index-3 subgroups when p > 3.<br />
<br />
2e. (extra credit) What can you say about the minimal index of a proper subgroup of SL_2(Z/pZ)?<br />
<br />
3. Let H be a subgroup of G of index 2. Prove that H is normal.<br />
<br />
4. We proved two theorems in class: Lagrange's theorem, which says that the order of a subgroup of G divides |G|, and Cauchy's theorem, which says that if p divides |G| then there exists a subgroup of G of order p. Cauchy's theorem is a kind of partial converse to Lagrange's theorem, but the full converse is false. Give an example of a finite group G and a divisor n of |G| such that you can prove there is no subgroup of G of order n.<br />
<br />
5. Let X be the set of lines in F_3^2 (here a line means a 1-dimensional linear subspace) so that |X| = 4. Let G be the projective linear group PGL_2(F_3), so that G acts on X via its action on F_3^2. This action can be thought of as a homomorphism from G to S_4. Show that this homomorphism is an ISOMORPHISM from PGL_2(F_3) to S_4.<br />
<br />
6. The group S_3 has only three conjugacy classes. Prove that a finite group with at most three conjugacy classes has order at most 6.<br />
<br />
7. Let X be the set of ordered triples of elements of {1,..,n}, for some n >= 3. Then S_n acts on X. How many orbits are there? (Hint: the answer does not depend on n.)<br />
<br />
8. Suppose that G and H are groups and f: G -> H is a homomorphism. Recall the definition of the abelianization G^ab from the previous problem set.<br />
<br />
8a. Show that the composition G -> H -> H^ab factors through a unique homomorphism G^ab -> H^ab, which we denote f^ab.<br />
<br />
8b. Show that if f: G -> H and g: H -> Q are homomorphisms, then f^ab o g^ab = (f o g)^ab.<br />
<br />
(For those reading MacLane, this constitutes a proof that abelianization is a '''functor''' from the category of groups to the category of abelian groups.)<br />
<br />
==HOMEWORK 5 (due Oct 23)==<br />
<br />
1. Show that there are only two different isomorphism classes of groups of order 14. (Hint: prove that such a group must be a semidirect product of a group of order 7 by a group of order 2.)<br />
<br />
2. (variant of Burnside's Lemma) Let G be a finite group and H a normal subgroup. Suppose G acts on a finite set X. <br />
<br />
2a. Show that if x and y lie in the same H-orbit, and g is an element of G, then gx and gy also lie in the same H-orbit. Thus, G/H has a natural action on the set of H-orbits X/H.<br />
<br />
2b. Show that the average number of fixed points in X of an element of the coset gH is the same as the number of fixed points of the element gH of G/H in its action on X/H.<br />
<br />
3. A ''central extension'' of a group G by an abelian group A is a group E, together with a surjective homomorphism E -> G whose kernel is central in E and is isomorphic to A.<br />
<br />
3a. Show that the quaternion group is a central extension of (Z/2Z)^2 by Z/2Z.<br />
<br />
3b. Give an example of another nonabelian group which is a central extension of (Z/2Z)^2 by Z/2Z and which is not isomorphic to the quaternion group.<br />
<br />
4. Prove that a central extension of an abelian group is nilpotent.<br />
<br />
5. Give two different composition series for S_4 and show that they have the same composition factors.<br />
<br />
6. Suppose that A is a finitely generated abelian group, and B is a subgroup of A.<br />
<br />
6a. Prove that B is finitely generated and that rank(B) <= rank(A).<br />
<br />
6b. Prove that rank(A/B) = rank(A) - rank(B).<br />
<br />
7. For this problem, you may use the fact (which I should have asked on homework earlier) that if H_1 and H_2 are subgroups of G, then<br />
<br />
[G: H_1 intersect H_2] <= [G:H_1][G:H_2].<br />
<br />
Prove that if G is a simple group, d is an integer greater than 1, and S is a subgroup of index d in G, then |G| <= d^d.<br />
<br />
Conclude that, in particular, A_5 has no subgroup of index 3. On the other hand, exhibit a subgroup of A_5 of index 5.<br />
<br />
(Note: d^d is not a sharp bound; can you get a bound of d factorial instead?)<br />
<br />
<!--(This argument shows that a solvable subgroup of A_n has size at most about (n-1)!, but this is far from sharp; e.g. I found a citation to a 1967 paper of Dixon which shows that a solvable subgroup of A_n has size at most k^n where k is around 2.88.)--><br />
<br />
<!--<br />
8. Give an example of an abelian group A with a subgroup B such that B is not a direct summand of A.<br />
<br />
9. Let A be the subgroup of Z^2 generated by the vector (1,0). Let B be the subgroup of Z^2 generated by the vector (3,3). Show that A and B are isomorphic to each other, but Z^2 / A is not isomorphic to Z^2 / B.<br />
<br />
==HOMEWORK 6 (due Oct 30)==<br />
<br />
1. In this problem, you will construct a functor F from the category of finite sets to the category of complex vector spaces. I'll tell you what it does on objects: if X is a finite set, then F(X) is the vector space spanned by a set of basis elements {e_x}_{x in X}, so that dim F(X) = |X|. Your job: complete the definition of F by describing the map from Mor_{Finite Sets}(X,Y) to Mor_{Vector Spaces}(F(X),F(Y)), and showing that your map respects composition.<br />
<br />
2. If (V, rho) is a representation of a group G, denote by V^G the space of vectors in V such that gv = v for all g in G; these are called _invariant_ vectors. We saw that the permutation representation of S_3 has a one-dimensional space of invariants. Prove that if X is a finite set with G-action, and V_X the corresponding permutation representation of G, then<br />
<br />
dim V_X^G = number of orbits of X.<br />
<br />
3. (Invariants are functorial) Let G be a group. Then there is a category called C[G]-Mod whose objects are complex representations of G (i.e. pairs (V,rho) where V is a complex vector space and rho is a homomorphism from G to GL(V)). The morphisms in this category are the ones described in class: Mor((V,rho),(W,psi)) is the set of linear maps f from V to W such that<br />
<br />
f(rho(g)(v)) = psi(g)(f(v))<br />
<br />
for all g in G and all v in V.<br />
<br />
Show that there is a functor H_0 from the category of representations of G to the category of vector spaces whose action on objects is given by<br />
<br />
H_0((V) = V^G.<br />
<br />
(In other words: your job again is to explain how to associate to a morphism from (V,rho) to (W,phi) a morphism from V^G to W^G, in a way that's compatible with composition.)<br />
<br />
<!--4. The group PSL_2(Z) has a presentation of the form [U,V|U^2 = V^3 = 1]. Thus, an n-dimensional representation of PSL_2(Z) is precisely a pair of n x n matrices (rho(U),rho(V)), such that rho(U)^2 = rho(V)^3 = 1. Prove that for every n > 1 there are uncountably many isomorphism classes of n-dimensional representations of PSL_2(Z). (Extra credit: show that there are uncountably many isomorphism classes of _irreducible_ representations.)<br />
<br />
An amazing theorem of Margulis (superrigidity) shows that SL_2(Z) is very special among arithmetic groups in this way. For instance, there are only _finitely_ many isomorphism classes of n-dimensional representations of PSL_d(Z) for any d > 2; those groups are "rigid."--><br />
<br />
<!--<br />
5. Let G be a finite group, let X be a set with G-action, and let chi be the character of the corresponding permutation representation. Show that, for each g in G, chi(g) is the number of elements of X fixed by g. (I am actually going to prove this in class.)<br />
<br />
6. Let V be the space of homogeneous degree-3 polynomials in variables x_1, x_2, x_3; then S_3 acts on V by permutation of the three variables. Describe the decomposition of V into irreducible representations of S_3.<br />
<br />
7a. Show that if chi is the character of a representation of a finite group G, then chi(g^{-1}) is the complex conjugate of chi.<br />
<br />
7b. As a consequence, show that if g is conjugate to g^{-1} for all g in G, then chi(g) is real.<br />
<br />
7c. The groups D_p and S_n have the property that g is conjugate to g^{-1} for _every_ g in G, so that every character is a real-valued function on G. Give an example of a group G where this is not the case, and give an example of a representation of G whose character takes non-real values.<br />
<br />
==HOMEWORK 7 (due Nov 6)==<br />
<br />
1. Let G be a finite group, let X be a finite set with G-action, and let chi be the character of the corresponding permutation representation. Show that, for each g in G, chi(g) is the number of elements of X fixed by g. (I am actually going to prove this in class.)<br />
<br />
2a. Show that if chi is the character of a finite-dimensional representation of a finite group G, then chi(g^{-1}) is the complex conjugate of chi(g) for all g.<br />
<br />
2b. As a consequence, show that if g is conjugate to g^{-1} for all g in G, then chi(g) is real.<br />
<br />
2c. The groups D_p and S_n have the property that g is conjugate to g^{-1} for _every_ g in G, so that every character is a real-valued function on G. Give an example of a group G where this is not the case, and give an example of a representation of G whose character takes non-real values.<br />
<br />
3. Let V and W be complex vector spaces. Let f be a linear transformation of V, and g a linear transformation of W. <br />
<br />
3a. Show that there is a unique linear transformation F satisfying<br />
<br />
F(v tensor w) = f(v) tensor g(w)<br />
<br />
for all v in V and all w in W. We denote this transformation by f tensor g.<br />
<br />
3b. Suppose V and W are finite dimensional. Show that Trace(f tensor g) = Trace(f) Trace(g). What does this say when f and g are both the identity transformation?<br />
<br />
4. Let V be a vector space. We define Sym^2 V to be the quotient of V tensor V by the subspace generated by all elements of the form<br />
<br />
v tensor w - w tensor v<br />
<br />
for v,w in V.<br />
<br />
Suppose dim V = n. What is dim Sym^2 V?<br />
<br />
5. Show that Z/pZ tensor Z/qZ is zero when p and q are distinct primes.<br />
<br />
==HOMEWORK 8 (due Nov 13)==<br />
<br />
1. (Schur's lemma can be false over non-algebraically closed fields) Let G be Z/3Z and let g be a generator of G. Then G acts by cyclic permutations on the set {1,2,3}; let P be the corresponding permutation representation on the REAL (not complex!) vector space generated by e_1,e_2,e_3.<br />
<br />
1a. Show that P breaks up as the direct sum of the trivial representation and a 2-dimensional representation V which is irreducible. (Remember, we are not working over the complex numbers, so you cannot use the criterion for irreducibility in terms of the character that we worked out in class for the complex case; you need to prove directly that V has no nontrivial proper subrepresentation.)<br />
<br />
1b. Show that the map from V to V induced by g is G-equivariant, but is neither zero nor a scalar multiplication.<br />
<br />
2. If V_1 and V_2 are representations (not necessarily irreducible!) of G, show that <chi_V_1, chi V_2> is a non-negative integer.<br />
<br />
3. When a formula always yields a non-negative integer, it is often because it is secretly a formula for either the cardinality of some finite set or the dimension of some finite-dimensional vector space. In this case, show that<br />
<br />
<chi_{V_1}, chi_{V_2}> = dim_C Hom_G(V_1,V_2)<br />
<br />
where we recall that Hom_G(V_1,G_2) refers to the space of G-equivariant homomorphisms from V_1 to V_2.<br />
<br />
4. If A is an abelian group, show that every irreducible representation of A is 1-dimensional. (Hint: show that there is a vector which is an eigenvector for every element of A!)<br />
<br />
5. Let G be the group of order 20 which is the semidirect product of A by H, where A is Z/5Z and H is (Z/5Z)^* with its natural action on A. We will work out all the irreducible representations of A. <br />
<br />
5a. First of all, consider the 5-dimensional permutation representation of G acting by left multiplication on the cosets of H. Show that this representation decomposes as a direct sum of the trivial representation and an irreducible 4-dimensional representation. (Hint: it is probably easier to use the character of this representation than to prove irreducibility directly.) <br />
<br />
5b. Now show that there are 4 1-dimensional representations of G which factor through the quotient G/A (i.e. such that rho(A) is the identity.)<br />
<br />
5c. Using the decomposition of the regular representation, show that the irreducible representations of G you have constructed are the only ones.<br />
<br />
==HOMEWORK 9 (due Nov 20)==<br />
<br />
0. Find all ring homomorphisms from Z to Z/30Z. [Warning: there are 8 of them, not 30.]<br />
<br />
<br />
1. If R is a ring (not necessarily with 1) such that a^2 = a for every a in R, R is called a Boolean ring.<br />
<br />
1a. Show that any nonzero Boolean ring has characteristic 2. (In other words, show that a+a=0 for all a in R.)<br />
<br />
1b. Show that every Boolean ring is commutative.<br />
<br />
<br />
2. Let phi : R -> S be a homomorphism of commutative rings with 1.<br />
<br />
2a. If I is an ideal of S, show that phi^(-1) (I) is an ideal of R.<br />
<br />
2b. If P is a prime ideal of S, show that phi^(-1) (P) is either R or a prime ideal of R.<br />
<br />
2c. If M is a maximal ideal of S and phi is surjective, show that phi^(-1) (M) is a maximal ideal of R.<br />
<br />
2d. Find a homomorphism phi : R -> S and a maximal ideal M of S such that phi^(-1) (M) is not a maximal ideal of R.<br />
<br />
<br />
3. Let Nil(A) denote the set of nilpotent elements of a ring A, and let R be a commutative ring. In class it was observed that Nil(R) is an ideal of R.<br />
<br />
3a. Find (a generator of) Nil(Z/720Z).<br />
<br />
3b. Show that Nil(R/Nil(R)) = 0.<br />
<br />
3c. If x is in Nil(R) and u is a unit, show that u+x is a unit.<br />
<br />
3d. Show that Nil(S), where S is the 2x2 matrices with real coefficients, is not an ideal of S.<br />
<br />
<br />
4a. If M is an R-module and I is an ideal of R such that im = 0 for all i in I and m in M, show that M carries the structure of an R/I-module.<br />
<br />
4b. If S is a commutative ring with 1 and J is a maximal ideal of S, show that J/J^2 is a vector space over S/J.<br />
<br />
<br />
5. A "divisible element" x of a Z-module is one such that for each positive integer k, there exists a y such that ky=x. Show that a free Z-module cannot contain a nonzero divisible element.<br />
<br />
<br />
6. Prove that the direct sum of any collection of free R-modules is also a free R-module.<br />
<br />
<br />
7. In this problem you will prove that the module M given by the direct product of copies of Z, indexed by the positive integers, is not a free Z-module. Therefore, assume that M is a free Z-module with basis B, and let N be the direct sum of copies of Z, as a submodule of M.<br />
<br />
7a. Show there is a countable subset B1 of B such that N is contained in the submodule N1 generated by B1. [Hint: N is countable.]<br />
<br />
7b. Show that M/N1 is a free Z-module.<br />
<br />
7c. Let S = {(b1, b2, b3, ...) : b_i = i! or -i!}. Show that there is some s in S which is not in N1. [Hint: S is uncountable.]<br />
<br />
7d. For any s in S but not N1, show that for each positive integer k, there exists some m in M with s-km in N1. Conclude that s+N1 is a divisible element of M/N1.<br />
<br />
7e. Use problem 5 to obtain a contradiction.<br />
<br />
<br />
8. In class I said "rank" is not always well-defined for R-modules if R is not commutative. Here is an example demonstrating this assertion. Let M be the countably infinite direct product of copies of Z as in exercise 7, and let R be the endomorphism ring of M. Define phi_1(a1,a2,a3,a4,...) = (a1,a3,a5,...) and phi_2(a1,a2,a3,a4,...) = (a2,a4,a6,...).<br />
<br />
8a. Show that {phi_1, phi_2} is a free basis for R, as a left R-module. [Hint: let psi_1(a1,a2,a3,...) = (a1,0,a2,0,...) and psi_2(a1,a2,a3,...) = (0,a1,0,a2,...). Compute the products of the phi_i and psi_j in either order, and use the result to show that the phi_i are a basis.]<br />
<br />
8b. Show that R is isomorphic to R^n for every positive integer n.<br />
<br />
<br />
==HOMEWORK 10 (due Dec 4)==<br />
<br />
1. '''How algebraists do calculus'''. The ''ring of dual numbers'' is the ring R = C[e]/(e^2). We think of e as an infinitesimal number; that is, something so small that its square is 0. This algebraic notion of infinitesimal is the basis for the algebraic notion of differentiation, which works as follows. Let f be a polynomial in R[x]. Show that there is a unique polynomial g in C[x] such that<br />
<br />
f(x+e) - f(x) = eg(x)<br />
<br />
and that in fact g is the derivative of f. In other words, the usual definition of "derivative" works just fine in this context, without any use of the notion of limit!<br />
<br />
2. Write down all the ideals of C[e]/e^2. (There are three.)<br />
<br />
3. Show that any right ideal of the ring M_2(Q) of 2x2 matrices is an ideal of the form I_V described in class for some subspace V of Q^2; that is, it consists precisely of those linear transformations whose image is contained in V.<br />
<br />
<br />
4. An ''idempotent'' element of a ring is an element e satisfying e^2 = e.<br />
<br />
4a. If e is an idempotent, show that 1-e is also an idempotent.<br />
<br />
4b. Suppose R has a unit (we are actually always supposing this, but I just want to remind you of it for this problem) and suppose e is a ''central'' idempotent, i.e. an idempotent which commutes with all elements of R. Show that eR and (1-e)R are both two-sided ideals. Moreover, show that eR and (1-e)R are also subrings of R (with the warning that the identity in eR is not 1, which is not contained in eR, but rather e.)<br />
<br />
4c. Show that the map f_e: R -> eR defined by f_e(r) = er is a ring homomorphism. Finally show that the ring homomorphism<br />
<br />
f_{e} x f_{1-e}: R -> eR x (1-e)R <br />
<br />
is actually an isomorphism of rings. In other words, the presence of a central idempotent induces a decomposition a ring as a direct product.<br />
<br />
<br />
5. The ''center'' of a ring R is the subring of elements commuting with every element of R.<br />
<br />
5a. Show that the center of M_n(Q) is the ring of scalar matrices.<br />
<br />
5b. Show that the center of the group ring C[G] is the set of elements sum f(g) g such that f(g) is a class function.<br />
<br />
<br />
6. '''How algebraists do differential equations.''' We define the ''Weyl algebra'' W of differential operators in one variable as follows. Multiplication by x is a linear transformation on the complex vector space V of all smooth functions on C. Differentiation, which we denote d, is also a linear transformation on V. So we let W be the subring of End(V) generated by x and d. We note that x and d satisfy the relation xd = dx - 1; in particular, the Weyl algebra is not commutative.<br />
<br />
6a. Describe the submodule of V which is annihilated by the element d-x in W.<br />
<br />
6b. Show that, for any polynomial f(x), fd-df = -f', and also show that for any polynomial g(d), gx-xg = -g'. (This is another way algebraists do calculus!) <br />
<br />
6c. Use the facts from (b) to show that the center of the Weyl algebra is just the field of constants. (Remark: you will use in the course of this proof the fact that a polynomial whose derivative is 0 is constant -- this would not be true if we were working in positive characteristic, as the example of x^p considered as a polynomial over F_p demonstrates.)<br />
<br />
<br />
7. If M is a left module for a ring R, we denote by Ann(M) the set of elements r in R such that rm = 0 for all m in M. Show that Ann(M) is a two-sided ideal. <br />
<br />
<br />
8. Let M be a left module for R, and let m be an element of M. We denote by Ann(m) the set of elements r in R such that rm = 0.<br />
<br />
8a. Show that Ann(m) is a left ideal, and show that the submodule Rm of M generated by m is isomorphic to the left module R/Ann(m).<br />
<br />
8c. Let V,W be as in question 6. Show that Ann(e^x) is the left ideal W(d-1).<br />
<br />
==HOMEWORK 11 (due Dec 13)==<br />
<br />
1. Let f: R -> S be a homomorphism of rings. As discussed in class, if M is a left R-module, then S tensor_R M is a left S-module, and if N is an S-module, then N can also be considered as an R-module by pullback of the action.<br />
<br />
Prove that if M is a finitely generated left R-module, then S tensor_R M is a finitely generated left S-module. On the other hand, give an example to show that N can be finitely generated as a left S-module but not finitely generated when considered as a left R-module.<br />
<br />
2. Recall from class that when V and W are complex vector spaces, W^dual tensor V is naturally isomorphic to the space of linear homomorphisms Hom(W,V).<br />
<br />
Now suppose that V and W are left C[G]-modules, i.e. complex representations of a group G. Show that W^dual tensor_C[G} V is the maximal G-invariant quotient of Hom(W,V); in other words, the quotient of Hom(W,V) by the subspace generated by all functions F of the form F(w) = f(gw) - gf(w) for some function f in Hom(W,V) and some g in G.<br />
<br />
3. '''How algebraists do logic.''' Why are Boolean rings called Boolean rings? Because they model some features of the Boolean arithmetic that governs propositional logic. Let A be a collection of propositions. We define two operations on A as follows. By a + b we mean the negation of a xor b, while by ab we mean "a or b."<br />
<br />
3a. It is not a priori obvious that these operations satisfy the ring axioms, but assume this for the moment. Show that a^2 = a for all a (so A is a boolean ring) and 2a = 0 for all a (so A has characteristic 2.)<br />
<br />
3b. Suppose we have a consistent assignment of truth-values to the propositions in A. Show that the set of true statements of A forms an ideal of A.<br />
<br />
3c. (Optional for people who like propositional logic) Show that a(b+c) = ab + ac with the given operations.<br />
<br />
<br />
4. Let K be Q[x]/(x^3-2).<br />
<br />
4a. Show that K is a field.<br />
<br />
4b. Show that K tensor_Q K decomposes as the direct sum of two fields. What are they? [Note: K tensor_Q K is a ring!]<br />
<br />
5. There was some discussion in class about when we can "go back and forth" between right modules and left modules, as is the case with group rings.<br />
<br />
5a. If R is a ring, we denote R^op ("the opposite ring") to be the ring whose elements are those of R, but whose multiplication law x_opp is defined by s x_opp r = rs. Show that if M is a left R-module, then the map M x R^op -> M sending (x,r) to rx makes M into a right R^op module. In particular, if there is a RING ISOMORPHISM from R to R^op, then every left R-module can be thought of as a right R-module.<br />
<br />
5b. Verify that the map sending g to g^{-1} extends by linearity to give a ring isomorphism between C[G] and C[G]^op.<br />
<br />
5c. Construct a ring isomorphism between M_n(Q) and M_n(Q)^op.<br />
<br />
5d. Optional: Give an example of a ring (with 1) which is not isomorphic to its opposite ring. (This is not so easy, I think! I found a MathOverflow thread with several examples, none of which came with a short proof of the non-isomorphism -- try your hand!)<br />
<br />
6. . Let R be a commutative ring with 1. If R has a unique maximal ideal M, R is called a "local ring".<br />
<br />
6a. If R is a local ring, show that every element of R that is not in M is a unit.<br />
<br />
6b. Conversely, if R is such that the set of nonunits forms an ideal I, show that R is a local ring with maximal ideal I.<br />
<br />
7. The inclusion of S_3 into S_4 associated to the inclusion of {1,2,3} into {1,2,3,4} induces a ring homomorphism from C[S_3} to C{S_4]. Let V be a representation of C{S_3]. Then, as discussed in class, V tensor_C[S_3] C[S_4] is a representation of S_4, called the '''induced representation''' of V.<br />
<br />
7a. Show that dim_C (V tensor_C[S_3] C[S_4]) = 4 dim_C V.<br />
<br />
7b. When V is the trivial representation of S_3, decompose (V tensor_C[S_3] C[S_4]) as a sum of irreducible representations of S_4.<br />
<br />
--><br />
<br />
<!--==HOMEWORK PROBLEMS TO GIVE IN FUTURE SEMESTERS==<br />
<br />
Let f: C[x] tensor_C C[y] -> C[z] be the map of rings with f(x) = f(y) = z. Show that f(m tensor n) is not 0 for any pure tensor m tensor n, but f is not injective. (This is a warning not to try to check injectivity by checking that no pure tensors are killed by the map.)<br />
<br />
--></div>Dummithttps://hilbert.math.wisc.edu/wiki/index.php?title=741&diff=62527412013-11-09T01:27:51Z<p>Dummit: /* HOMEWORK 9 (Due Nov 13) */</p>
<hr />
<div>'''Math 741'''<br />
<br />
Algebra<br />
<br />
Prof: [http://www.math.wisc.edu/~ellenber Jordan Ellenberg]<br />
<br />
Grader: Evan Dummit.<br />
* Homework policies:<br />
* Late homework may be given directly to the grader, along with either <br />
(i) the instructor's permission, or (ii) a polite request for mercy.<br />
* Assignments that are more than 1 page should be affixed in some reasonable way.<br />
* Results from places (e.g., the internet) other than 741 and standard books must be cited.<br />
<br />
JE's office hours: Monday 12pm-1pm (right after class)<br />
<br />
This course, the first semester of the introductory graduate sequence in algebra, will cover the basic theory of groups, group actions, representation, linear and multilinear algebra, and the beginnings of ring theory.<br />
<br />
==SYLLABUS==<br />
<br />
In this space we will record the theorems and definitions we covered each week, which we can use as a list of notions you should be prepared to answer questions about on the Algebra qualifying exam. The material covered on the homework is also an excellent guide to the scope of the course. <br />
<br />
'''WEEK 1''': <br />
<br />
Definition of group. Associativity. Inverse. <br />
<br />
Examples of group: GL_n(R). GL_n(Z). Z/nZ. R. Z. R^*. The free group F_k on k generators. <br />
<br />
Homomorphisms. The homomorphisms from F_k to G are in bijection with G^k. Isomorphisms.<br />
<br />
'''WEEK 2''':<br />
<br />
The symmetric group (or permutation group) S_n on n letters. Cycle decomposition of a permutation. Order of a permutation. Thm: every element of a finite group has finite order. <br />
<br />
Subgroups. Left and right cosets. Lagrange's Theorem. Cyclic groups. The order of an element of a finite group is a divisor of the order of the group.<br />
<br />
The sign homomorphism S_n -> +-1.<br />
<br />
<br />
'''WEEK 3'''<br />
<br />
Normal subgroups. The quotient of a group by a normal subgroup. The first isomorphism theorem. Examples of S_n -> +-1 and S_4 -> S_3 with kernel V_4, the Klein 4-group.<br />
<br />
Centralizers and centers. Abelian groups. The center of SL_n(R) is either 1 or +-1.<br />
<br />
Groups with presentations. The infinite dihedral group <x,y | x^2 = 1, y^2 = 1>.<br />
<br />
'''WEEK 4'''<br />
<br />
More on groups with presentations.<br />
<br />
Second and third isomorphism theorems.<br />
<br />
Semidirect products.<br />
<br />
'''WEEK 5'''<br />
<br />
Group actions, orbits, and stabilizers.<br />
<br />
Orbit-stabilizer theorem.<br />
<br />
Cayley's theorem.<br />
<br />
Cauchy's theorem.<br />
<br />
'''WEEK 6'''<br />
<br />
Applications of orbit-stabilizer theorem (p-groups have nontrivial center, first Sylow theorem.)<br />
<br />
Classification of finite abelian groups and finitely generated abelian groups.<br />
<br />
Composition series and the Jordan-Holder theorem (which we state but don't prove.) <br />
<br />
The difference between knowing the composition factors and knowing the group (e.g. all p-groups of the same order have the same composition factors.)<br />
<br />
'''WEEK 7'''<br />
<br />
Simplicity of A_n.<br />
<br />
Nilpotent groups (main example: the Heisenberg group)<br />
<br />
Derived series and lower central series.<br />
<br />
Category theory 101: Definition of category and functor. Some examples. A group is a groupoid with one object.<br />
<br />
'''WEEK 8'''<br />
<br />
Introduction to representation theory.<br />
<br />
'''WEEK 10'''<br />
<br />
Ring theory 101: Rings, ring homomorphisms, ideals, isomorphism theorems. Examples: fields, Z, the Hamilton quaternions, matrix rings, rings of polynomials and formal power series, quadratic integer rings, group rings. Integral domains. Maximal and prime ideals. The nilradical.<br />
<br />
Module theory 101: Modules, module homomorphisms, submodules, isomorphism theorems. Noetherian modules and Zorn's lemma. Direct sums and direct products of arbitrary collections of modules.<br />
<br />
<br />
==HOMEWORK 1 (due Sep 16)==<br />
<br />
1. Suppose that H_1 and H_2 are subgroups of a group G. Prove that the intersection of H_1 and H_2 is a subgroup of G.<br />
<br />
2. Recall that S_3 (the symmetric group) is the group of permutations of the set {1..3}. List all the subgroups of S_3.<br />
<br />
3. We can define an equivalence relation on rational numbers by declaring two rational numbers to be equal whenever they differ by an integer. We denote the set of equivalence classes by Q/Z. The operation of addition makes Q/Z into a group.<br />
<br />
a) For each n, prove that Q/Z has a subgroup of order n.<br />
<br />
b) Prove that Q/Z is a ''divisible'' group: that is, if x is an element of Q/Z and n is an integer, there exists an element y of Q/Z such that ny = x. (Note that we write the operation in this group as addition rather than multiplication, which is why we write ny for the n-fold product of y with itself rather than y^n)<br />
<br />
c) Prove that Q/Z is not finitely generated. (Hint: prove that if x_1, .. x_d is a finite subset of Q/Z, the subgroup of Q/Z generated by x_1, ... x_d is finite.)<br />
<br />
d) Conclude that Q is not finitely generated.<br />
<br />
4. We will prove that there is no homomorphism from SL_2(Z) to Z except the one which sends all of SL_2(Z) to 0. Suppose f is a homomorphism from SL_2(Z) to Z.<br />
<br />
a) Let U1 be the upper triangular matrix with 1's on the diagonal and a 1 in the upper right hand corner, as in class, and let U2 be the transpose of U1, also as in class. Show that (U1 U2^{-1})^6 = identity (JING 1, TAO 0) and explain why this implies that f(U1) = f(U2).<br />
<br />
b) Show that there is a matrix A in SL_2(Z) such that A U1 A^{-1} = U2^{-1}. (Recall that we say U1 and U2^{-1} are "conjugate".) Explain why this also implies that f(U1) = -f(U2).<br />
<br />
c) Explain why a) and b) imply that f must be identically 0.<br />
<br />
5. The argument above also shows that there is no nonzero homomorphism from SL_2(Z) to Z/pZ where p is a prime greater than 3. However, it leaves open the possibility that there is indeed a nonzero homomorphism from SL_2(Z) to Z/2Z. Exhibit such a homomorphism. Optional challenge problem: exhibit a nonzero homomorphism from SL_2(Z) to Z/3Z.<br />
<br />
<br />
==HOMEWORK 2 (due Sep 25)==<br />
<br />
1. Let V be the Klein 4-group in S_4. Let Q be the symmetric group on the set DF of double-flips in S_4; there are 3 double flips, so Q is isomorphic to S_3. If g is an element of S_4, we discussed in class that conjugation by g permutes the elements of DF. So to each element of g, we have associated an element f(g) of Q. More or less by definition, this defines a homomorphism f: G -> Q. Show that this homomorphism is surjective and has kernel H, and thus that G/H is isomorphic to Q.<br />
<br />
2. Let F_2 be the free group on two generators, which we denote x,y. Prove that there is a automorphism of F_2 which sends x to xyx and y to xy, and prove that this automorphism is unique.<br />
<br />
3. Let H be a subgroup of G, and let N_G(H), the ''normalizer'' of H in G, be the set of elements g in G satisfying g H g^{-1} = H. Prove that N_G(H) is a subgroup of G.<br />
<br />
4. Let T be the subgroup of GL_2(R) consisting of diagonal matrices. This is an example of a "Cartan subgroup," or "torus" (whence the notation T.) Describe the normalizer N(T) of T in GL_2(R). Prove that there is a homomorphism from N(T) to Z/2Z whose kernel is precisely T. <br />
<br />
5. Prove that the group of inner automorphisms of a group is normal in the full automorphism group Aut(G).<br />
<br />
6. Let H and H' be subgroups of a group G, and let x be an element of G. We denote by HxH' the set of elements of the form hxh', with h in H and h' in H', and we call HxH a ''double coset'' of the pair (H,H').<br />
<br />
a) Show that G decomposes as a disjoint union of double cosets of (H,H').<br />
<br />
b) Let G be GL_2(R) and let B be the subgroup of upper triangular matrices. Prove that G decomposes into exactly two double cosets of (B,B). (This is an example of the so-called *Bruhat decomposition* which is of great importance in the theory of algebraic groups and their representations.)<br />
<br />
c) Let G be the symmetric group S_n, and let H be the subgroup of permutations which fix 1. Describe the double coset decomposition of S_n into double cosets of (H,H).<br />
<br />
<br />
==HOMEWORK 3 (due Oct 2)==<br />
<br />
1. Compute the centralizer of the triple flip (12)(34)(56) in S_6. (Note: this is the kind of problem that a computer algebra package like sage is very good at, and I encourage you to learn to use sage well enough to check your answer.)<br />
<br />
2. Warning: a normal subgroup of a normal subgroup need not be normal! Let H be the subgroup of S_4 generated by (12)(34). Then H is a subgroup of the Klein 4-group V_4, which we have already shown is normal in S_4. Show that H is normal in V_4, but H is not normal in S_4.<br />
<br />
3. Let G be a group and let H_1 and H_2 be subgroups of G of finite index (that is, the number of cosets in G/H_i is finite, i = 1,2.) Prove that H_1 intersect H_2 also has finite index in G.<br />
<br />
4. Let X be the set of lines in F_3^2 (here a line means a 1-dimensional linear subspace) so that |X| = 4. Let G be the projective linear group PGL_2(F_3), so that G acts on X via its action on F_3^2. This action can be thought of as a homomorphism from G to S_4. Show that this homomorphism is an ISOMORPHISM from PGL_2(F_3) to S_4.<br />
<br />
5. We proved two theorems in class: Lagrange's theorem, which says that the order of a subgroup of G divides |G|, and Cauchy's theorem, which says that if p divides |G| then there exists a subgroup of G of order p. Cauchy's theorem is a kind of partial converse to Lagrange's theorem, but the full converse is false. Give an example of a finite group G and a divisor n of |G| such that you can prove there is no subgroup of G of order n.<br />
<br />
6. The group S_3 has only three conjugacy classes. Prove that a finite group with at most three conjugacy classes has order at most 6.<br />
<br />
7. Let X be the set of ordered triples of elements of {1,..,n}, for some n >= 3. Then S_n acts on X. How many orbits are there? (Hint: the answer does not depend on n.) OPTIONAL: How many orbits are there on the space of ordered k-tuples, when n >= k? <br />
<br />
==HOMEWORK 4 (due Oct 9)==<br />
<br />
1. Recall that the commutator [x,y] is x y x^{-1} y^{-1}. If G is a group, the commutator subgroup of G, denoted G' or [G,G], is the subgroup of G generated by all commutators [x,y] for x,y in G.<br />
<br />
1a. Show that G' is a normal subgroup of G.<br />
1b. Show that G/G' is an abelian group, which we call the abelianization, G^ab.<br />
1c. Show that if f: G -> A is a homomorphism, with A abelian, then G' is contained in ker f. Conclude that f factors through a homomorphism G/G' -> A.<br />
1d. Show that G' = G if and only if G has no nontrivial abelian quotient. In this case, we say G is ''perfect''.<br />
<br />
2. Prove that any p-Sylow subgroup of GL_n(F_p) is contained in SL_n(F_p).<br />
<br />
3. Let U_n(F_p) be the subgroup of GL_n(F_p) consisting of upper triangular matrices with 1's on the diagonal. Show that U_n(F_p) is a p-Sylow subgroup of GL_n(F_p).<br />
<br />
4. U_n(F_p) is called the *unipotent subgroup* of GL_n(F_p), because it has the property (not part of your homework, but easy to check) that every u in U_n(F_p) is unipotent, i.e. its characteristic polynomial is (x-1)^n. It is a p-group (as you know by virtue of the last problem) so it must have a nontrivial center. What is the center of U_nF_p)?<br />
<br />
5. U_3(F_p) is Shamgar Gurevich's favorite group, the *Heisenberg group" of order p^3. Write down a composition series for U_3(F_p).<br />
<br />
6. Let A be the subgroup of Z^2 generated by the vector (1,0). Let B be the subgroup of Z^2 generated by the vector (3,3). Show that A and B are isomorphic to each other, but Z^2 / A is not isomorphic to Z^2 / B.<br />
<br />
7. Suppose that G and H are groups and f: G -> H is a homomorphism. Recall the definition of the abelianization G^ab from problem 1.<br />
<br />
7a. Show that the composition G -> H -> H^ab factors through a unique homomorphism G^ab -> H^ab, which we denote f^ab.<br />
7b. Show that if f: G -> H and g: H -> Q are homomorphisms, then f^ab o g^ab = (f o g)^ab.<br />
<br />
(For those reading MacLane, this constitutes a proof that abelianization is a '''functor''' from the category of groups to the category of abelian groups.)<br />
<br />
8. Give two different composition series for S_4 and show that they have the same composition factors.<br />
<br />
OPTIONAL: We saw in class that a finitely generated abelian group that was torsion-free was in fact a free abelian group. This totally dies without the hypothesis of abelianness. To see this, give an example of a finitely generated group which is torsion free but which is not a free group, and which does not even *contain* any free group of rank greater than 1.<br />
<br />
==HOMEWORK 5 (due Oct 16)==<br />
<br />
1. Let G be a group, thought of as a category with one object. Show that a set with an action of G is the same thing as a functor from G to the category of sets.<br />
<br />
2. If (V, rho) is a representation of a group G, denote by V^G the space of vectors in V such that gv = v for all g in G; these are called _invariant_ vectors. We saw that the permutation representation of S_3 has a one-dimensional space of invariants. Prove that if X is a finite set with G-action, and V_X the corresponding permutation representation of G, then<br />
<br />
dim V_X^G = number of orbits of X.<br />
<br />
3. (Invariants are functorial) Let G be a group. Then there is a category called C[G]-Mod whose objects are complex representations of G (i.e. pairs (V,rho) where V is a complex vector space and rho is a homomorphism from G to GL(V)). The morphisms in this category are the ones described in class: Mor((V,rho),(W,psi)) is the set of linear maps f from V to W such that<br />
<br />
f(rho(g)(v)) = psi(g)(f(v))<br />
<br />
for all g in G and all v in V.<br />
<br />
Show that there is a functor H_0 from the category of representations of G to the category of vector spaces whose action on objects is given by<br />
<br />
H_0((V) = V^G.<br />
<br />
(In other words: your job again is to explain how to associate to a morphism from (V,rho) to (W,phi) a morphism from V^G to W^G, in a way that's compatible with composition.)<br />
<br />
4. Let G be a finite group, let X be a set with G-action, and let chi be the character of the corresponding permutation representation. Show that, for each g in G, chi(g) is the number of elements of X fixed by g. (I am actually going to prove this in class.)<br />
<br />
5. Let V be the space of homogeneous degree-3 polynomials in variables x_1, x_2, x_3; then S_3 acts on V by permutation of the three variables. Describe the decomposition of V into irreducible representations of S_3.<br />
<br />
6a. Show that if chi is the character of a representation of a finite group G, then chi(g^{-1}) is the complex conjugate of chi.<br />
<br />
6b. As a consequence, show that if g is conjugate to g^{-1} for all g in G, then chi(g) is real.<br />
<br />
6c. The groups D_p and S_n have the property that g is conjugate to g^{-1} for _every_ g in G, so that every character is a real-valued function on G. Give an example of a group G where this is not the case, and give an example of a representation of G whose character takes non-real values.<br />
<br />
OPTIONAL: (for people who know some topology.) Let $X$ be a topological space and let $Paths/X$ be the category of paths up to isotopy as described in class. Explain how to construct a bijection between {functors from Paths/X to the category FinSet of finite sets} and {finite covering spaces of X}. (For simplicity you may assume X is path-connected.)<br />
<br />
==HOMEWORK 6 (due Oct 23)==<br />
<br />
1. Let V_k be the space of homogenous degree-k polynomials in x_1, x_2, x_3. Compute the character of S_3 acting on V_k, and, using this, give formulas for the multiplicity of the trivial rep, the sign rep, and the standard rep of S_3 in V_k.<br />
<br />
2. (Schur's lemma can be false over non-algebraically closed fields) Let G be Z/3Z and let g be a generator of G. Then G acts by cyclic permutations on the set {1,2,3}; let P be the corresponding permutation representation on the REAL (not complex!) vector space generated by e_1,e_2,e_3.<br />
<br />
2a. Show that P breaks up as the direct sum of the trivial representation and a 2-dimensional representation V which is irreducible. (Remember, we are not working over the complex numbers, so you cannot use the criterion for irreducibility in terms of the character that we worked out in class for the complex case; you need to prove directly that V has no nontrivial proper subrepresentation.)<br />
<br />
2b. Show that the map from V to V induced by g is G-equivariant, but is neither zero nor a scalar multiplication.<br />
<br />
<br />
The following two problems involve the tensor product. I am going to teach the general notion of tensor product of modules in this course, but I am expecting you already know what the tensor product of vector spaces. If not, please review it by learning the definition and doing these problems, because having a familiarity with the vector space theory will be very helpful in learning the general story.<br />
<br />
3. Let V and W be complex vector spaces. Let f be a linear transformation of V, and g a linear transformation of W. <br />
<br />
3a. Show that there is a unique linear transformation F satisfying<br />
<br />
F(v tensor w) = f(v) tensor g(w)<br />
<br />
for all v in V and all w in W. We denote this transformation by f tensor g.<br />
<br />
3b. Suppose V and W are finite dimensional. Show that Trace(f tensor g) = Trace(f) Trace(g). What does this say when f and g are both the identity transformation?<br />
<br />
4. Let V be a vector space. We define Sym^2 V to be the quotient of V tensor V by the subspace generated by all elements of the form<br />
<br />
v tensor w - w tensor v<br />
<br />
for v,w in V.<br />
<br />
Suppose dim V = n. What is dim Sym^2 V?<br />
<br />
5. For any n, the group S_n has an (n-1)-dimensional representation, the "standard representation," on the space of vectors in C^n whose coordinates sum to 0. When n=3, we proved in class that this representation was irreducible. Prove that the standard representation is irreducible for all n.<br />
<br />
6. Give an example showing that the standard representation of S_n need NOT be irreducible over a field of characteristic p.<br />
<br />
7. Let H be a subgroup of G and let V be a representation of G. Let chi_{G/H} be the character of the permutation representation of G on the set of cosets G/H, so that, by problem 1, chi_{G/H}(g) is the number of fixed points of g in its action on the cosets. As discussed in class, the dimension of the G-invariant subspace V^G is given by the inner product of chi_V with the trivial character. Prove that the dimension of V^H is given by the inner product of chi_V with chi_{G/H}.<br />
<br />
==HOMEWORK 7 (due Oct 30)==<br />
<br />
1. When a formula always yields a non-negative integer, it is often because it is secretly a formula for either the cardinality of some finite set or the dimension of some finite-dimensional vector space. In this case, show that<br />
<br />
<chi_{V_1}, chi_{V_2}> = dim_C Hom_G(V_1,V_2)<br />
<br />
where we recall that Hom_G(V_1,G_2) refers to the space of G-equivariant homomorphisms from V_1 to V_2.<br />
<br />
2. Let G be the group of affine linear transformations of F_5; that is, it is the group of transformations x -> ax + b where a lies in (Z/5Z)^* and b lies in (Z/5Z). Note that G has order 20. Write H for the subgroup of G consisting of transformations fixing 0 (i.e. those for which b=0). <br />
<br />
We will work out all the irreducible representations of G. <br />
<br />
2a. First of all, consider the 5-dimensional permutation representation of G acting by left multiplication on the cosets of H. Show that this representation decomposes as a direct sum of the trivial representation and an irreducible 4-dimensional representation. (Hint: it is probably easier to use the character of this representation than to prove irreducibility directly.) <br />
<br />
2b. Now show that there are 4 1-dimensional representations of G.<br />
<br />
2c. Show that the irreducible representations of G you have constructed are the only ones.<br />
<br />
3. If G and H are finite groups, and V is a representation of G, and W is a representation of H, then V tensor W is a representation of GxH, and the character chi_{V tensor W} is given by<br />
<br />
chi_{V tensor W}(g,h) = chi_V(g) chi_W(h).<br />
<br />
3a. Prove that V tensor W is irreducible if and only if V and W are irreducible.<br />
<br />
3b. Prove that all the irreducible representations of G x H are of the form V tensor W.<br />
<br />
(Together, these propositions show that you can completely describe the representation theory of G x H in terms of that of G and that of H.)<br />
<br />
4. Suppose that V is an irreducible representation of S_n, and suppose that when we consider V as a representation of the alternating group A_n it is NOT irreducible. Prove that V, considered as a representation of A_n, is the direct sum of two non-isomorphic irreducible representations. Prove furthermore that chi_V(g) = 0 for all odd permutations g. Give an example of such an irreducible representation of S_4.<br />
<br />
<br />
OPTIONAL (because I couldn't quickly think of an easy way to do it!) Prove that there exists such a representation of S_n for every n >= 3.<br />
<br />
5. (Fourier analysis over finite fields.) I said in class that the representation theory that goes into Fourier analysis is different from the representation theory of finite groups we do in class, but that's not quite true; when you do Fourier theory over finite fields, the two theories come into much closer contact, with the bonus that we don't have to worry about issues of infinite sums that I hand-waved away in class.<br />
<br />
Let F be the field of p elements and let V be the space of complex-valued functions on F; so V is a p-dimensional space. Let G be the group Z/pZ (in other words, it is the additive group of the field.)<br />
<br />
Now V is a representation of G, in the same way we discussed in class: if f is a function in V, and a is an element of g, then the function gf is defined by<br />
<br />
gf(x) = f(x+a)<br />
<br />
5a. Describe the breakup of V into irreducible representations of G, which are all 1-dimensional (not just what the characters are, explicitly decribe the irreducible constituents of V as subspaces of V!)<br />
<br />
5b. There is a natural norm on V which sends f to ||f|| = sum_x |f(x)|^2.<br />
<br />
Given any f, we have a unique decomposition f = sum_i f_i, where f_i lies in the irreducible constituent V_i of V.<br />
<br />
Prove that ||f|| = sum_i ||f_i||.<br />
<br />
OPTIONAL (for people who are taking analysis) Explain why ||f_i|| is correctly thought of as a "Fourier coefficient" and why the fact proved in 5b is the finite-field analogue of Parseval's identity.<br />
<br />
==HOMEWORK 8 (due Nov 6)==<br />
<br />
1. If R is a ring (not necessarily with 1) such that a^2 = a for every a in R, R is called a Boolean ring.<br />
<br />
1a. Show that any nonzero Boolean ring has characteristic 2. (In other words, show that a+a=0 for all a in R.)<br />
<br />
1b. Show that every Boolean ring is commutative.<br />
<br />
2. Let phi : R -> S be a homomorphism of commutative rings (recalling that rings for us have a multiplicative identity 1 and that ring homomorphisms take 1 to 1.)<br />
<br />
2a. If I is an ideal of S, show that phi^(-1) (I) is an ideal of R.<br />
<br />
2b. If P is a prime ideal of S, show that phi^(-1) (P) is either R or a prime ideal of R.<br />
<br />
2c. If M is a maximal ideal of S and phi is surjective, show that phi^(-1) (M) is a maximal ideal of R.<br />
<br />
2d. Find a homomorphism phi : R -> S and a maximal ideal M of S such that phi^(-1) (M) is not a maximal ideal of R.<br />
<br />
3. Let Nil(R) denote the set of nilpotent elements of a ring R; that is, the set of x such that x^k = 0 for some positive integer k. <br />
<br />
3a. Show that Nil(R) is an ideal whenever R is a commutative ring. Assume R is commutative for the remainder of this exercise.<br />
<br />
3b. Describe Nil(Z/720Z).<br />
<br />
3c. If x is in Nil(R) and u is a unit, show that u+x is a unit.<br />
<br />
3d. Show that Nil(M_2(R)) is not an ideal of the matrix algebra M_2(R). (That is, it is neither a left nor a right ideal.)<br />
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4. We say a ring is Noetherian if any ascending chain of ideals I_1 < I_2 < I_3 < …. eventually stabilizes. This is a somewhat funny-looking condition if you haven't encountered it before, but it turns out to be a very useful way of formalizing the notion that a ring is "not too bad."<br />
<br />
4a. Show that Z is Noetherian.<br />
<br />
4b. Let R be the ring of continuous functions from the real numbers to the real numbers. Show that R is not Noetherian.<br />
<br />
5. '''How algebraists do calculus'''. The ''ring of dual numbers'' is the ring A = C[e]/(e^2). We think of e as an infinitesimal number; that is, something so small that its square is 0. This algebraic notion of infinitesimal is the basis for the algebraic notion of differentiation, which works as follows. Let f be a polynomial in C[x]. Show that there is a unique polynomial g in C[x] such that<br />
<br />
f(x+e) - f(x) = eg(x)<br />
<br />
and that in fact g is the derivative of f. In other words, the usual definition of "derivative" works just fine in this context, without any use of the notion of limit!<br />
<br />
6. Write down all the ideals of C[e]/e^2. (There are three.)<br />
<br />
7. Show that any right ideal of the ring M_2(Q) of 2x2 matrices is an ideal of the form I_V described in class for some subspace V of Q^2; that is, it consists precisely of those linear transformations whose image is contained in V.<br />
<br />
== HOMEWORK 9 (Due Nov 13) ==<br />
<br />
1. If M is an R-module and I is an ideal of R such that im = 0 for all i in I and m in M, show that M carries the structure of an R/I-module.<br />
<br />
2. It follows from 1. that if S is a commutative ring and J is a maximal ideal of S, that J/J^2 is a vector space over the field S/J. Compute the dimension of J/J^2 as a vector space over S/J when<br />
<br />
2a. S = Z, J = (5);<br />
2b. S = C[x,y], J = (x,y)<br />
<br />
2c. Suppose S = C[x,y]/(f) and J = (x,y). Then the dimension of J/J^2 depends on the choice of the polynomial f. Compute the possible values of this dimension and give an example of an f realizing each possibility. (NOTE: this is secretly, or not-so-secretly, another instance of differential geometry as carried out by algebraists...)<br />
<br />
3. An ''idempotent'' element of a ring is an element e satisfying e^2 = e.<br />
<br />
3a. If e is an idempotent, show that 1-e is also an idempotent.<br />
<br />
3b. Suppose R has a unit (we are actually always supposing this, but I just want to remind you of it for this problem) and suppose e is a ''central'' idempotent, i.e. an idempotent which commutes with all elements of R. Show that eR and (1-e)R are both two-sided ideals. Moreover, show that eR and (1-e)R are also subrings of R (with the warning that the identity in eR is not 1, which is not contained in eR, but rather e.)<br />
<br />
3c. Show that the map f_e: R -> eR defined by f_e(r) = er is a ring homomorphism. Finally show that the ring homomorphism<br />
<br />
f_{e} x f_{1-e}: R -> eR x (1-e)R <br />
<br />
is actually an isomorphism of rings. In other words, the presence of a central idempotent induces a decomposition a ring as a direct product.<br />
<br />
4. Let k be an algebraically closed field, and let D be a finite-dimensional division algebra over k. Prove that D = k. (Hint: if x is an element of D which is not in k, prove that there is some univariate polynomial f such that f(x) = 0. Then think back to our proof in class that the quaternion algebra with complex coefficients was not a division algebra.)<br />
<br />
5. The ''center'' of a ring R is the subring of elements commuting with every element of R.<br />
<br />
5a. Show that the center of M_n(Q) is the ring of scalar matrices.<br />
<br />
5b. Show that the center of the group ring C[G] is the set of elements sum f(g) g such that f(g) is a class function. Compute the central idempotents of C[S_3].<br />
<br />
6. '''How algebraists do differential equations.''' We define the ''Weyl algebra'' W of differential operators in one variable as follows. Multiplication by x is a linear transformation on the complex vector space V of all smooth functions on C. Differentiation, which we denote d, is also a linear transformation on V. So we let W be the subring of End(V) generated by x and d. We note that x and d satisfy the relation xd = dx - 1; in particular, the Weyl algebra is not commutative.<br />
<br />
6a. Describe the submodule of V which is annihilated by the element d-x in W.<br />
<br />
6b. Show that, for any polynomial f(x), fd-df = -f', and also show that for any polynomial g(d), gx-xg = -g'. (This is another way algebraists do calculus!) <br />
<br />
6c. Use the facts from 6b. to show that the center of the Weyl algebra is just the field of constants. (Remark: you will use in the course of this proof the fact that a polynomial whose derivative is 0 is constant -- this would not be true if we were working in positive characteristic, as the example of x^p considered as a polynomial over F_p demonstrates.)<br />
<br />
7. If M is a left module for a ring R, we denote by Ann(M) the set of elements r in R such that rm = 0 for all m in M. Show that Ann(M) is a two-sided ideal. <br />
<br />
8. Let M be a left module for R, and let m be an element of M. We denote by Ann(m) the set of elements r in R such that rm = 0.<br />
<br />
8a. Show that Ann(m) is a left ideal, and show that the submodule Rm of M generated by m is isomorphic to the left module R/Ann(m).<br />
<br />
8c. Let V,W be as in question 6. Show that Ann(e^x) is the left ideal W(d-1).<br />
<br />
<!--Below you will find a repository of homework problems. Note that some of these problems are taken from Lang's ''Algebra''.<br />
<br />
==HOMEWORK 1 (due Sep 20)==<br />
<br />
1. Suppose that H_1 and H_2 are subgroups of a group G. Prove that the intersection of H_1 and H_2 is a subgroup of G.<br />
<br />
2. Recall that S_3 (the symmetric group) is the group of permutations of the set {1..3}. List all the subgroups of S_3.<br />
<br />
3. We can define an equivalence relation on rational numbers by declaring two rational numbers to be equal whenever they differ by an integer. We denote the set of equivalence classes by Q/Z. The operation of addition makes Q/Z into a group.<br />
<br />
a) For each n, prove that Q/Z has a subgroup of order n.<br />
<br />
b) Prove that Q/Z is a ''divisible'' group: that is, if x is an element of Q/Z and n is an integer, there exists an element y of Q/Z such that ny = x. (Note that we write the operation in this group as addition rather than multiplication, which is why we write ny for the n-fold product of y with itself rather than y^n)<br />
<br />
c) Prove that Q/Z is not finitely generated. (Hint: prove that if x_1, .. x_d is a finite subset of Q/Z, the subgroup of Q/Z generated by x_1, ... x_d is finite.)<br />
<br />
4. Let F_2 be the free group on two generators x,y. Prove that there is a automorphism of F_2 which sends x to xyx and y to xy, and prove that this automorphism is unique.<br />
<br />
5. Let H be a subgroup of G, and let N_G(H), the ''normalizer'' of H in G, be the set of elements g in G satsifying g H g^{-1} = H. Prove that N_G(H) is a subgroup of G.<br />
<br />
6. Let T be the subgroup of GL_2(R) consisting of diagonal matrices. This is an example of a "Cartan subgroup," or "torus" (whence the notation T.) Describe the normalizer N(T) of T in GL_2(R). Prove that there is a homomorphism from N(T) to Z/2Z whose kernel is precisely T. <br />
<br />
7. Prove that the group of inner automorphisms of a group is normal in the full automorphism group Aut(G).<br />
<br />
8. Let H and H' be subgroups of a group G, and let x be an element of G. We denote by HxH' the set of elements of the form hxh', with h in H and h' in H', and we call HxH a ''double coset'' of the pair (H,H').<br />
<br />
a) Show that G decomposes as a disjoint union of double cosets of (H,H').<br />
<br />
b) Let G be GL_2(R) and let B be the subgroup of upper triangular matrices. Prove that G decomposes into exactly two double cosets of (B,B).<br />
<br />
c) Let G be the symmetric group S_n, and let H be the subgroup of permutations which fix 1. Describe the double coset decomposition of S_n into double cosets of (H,H).<br />
<br />
==HOMEWORK 2 (due Sep 27)==<br />
<br />
1. Prove that the symmetric group S_n is generated by transpositions. Hint: proceed by induction on n. Let H be the subgroup of S_n consisting of permutations fixing 1. Observe that H is isomorphic to S_{n-1}, so that H is generated by transpositions. Explain why it suffices to show that, for any g in S_n, the coset gH contains a transposition. Then explain why this is the case.<br />
<br />
2. Given that the symmetric group S_n is generated by transpositions, show that there are no nontrivial homomorphisms from S_n to Z/pZ for any odd prime p.<br />
<br />
3. Compute the centralizer of the triple flip (12)(34)(56) in S_6. (Note: this is the kind of problem that a computer algebra package like sage is very good at, and I encourage you to learn to use sage well enough to check your answer.)<br />
<br />
4. Warning: a normal subgroup of a normal subgroup need not be normal! Let H be the subgroup of S_4 generated by (12)(34). Then H is a subgroup of the Klein 4-group V_4, which we have already shown is normal in S_4. Show that H is normal in V_4, but H is not normal in S_4.<br />
<br />
5. The '''quaternion group''' Q is a non-abelian finite group of order 8. Its elements are 1, -1, i, -i, j, -j, k, -k; it satisfies g^2 = -1 for all g except +-1, and ijk = 1. (Note that the notation "-i" means "the product of i with -1" and -1 is understood to be central in Q.) Prove that every subgroup of Q is normal.<br />
<br />
6. (finishing example done in class) Let Gamma be the group F<x,y> / (x^2 = 1, y^2 = 1). (Remember, this means the quotient of the free group by the group generated by all conjugates of x^2 and y^2.) Prove that Gamma is infinite. Prove that Gamma has a subgroup R generated by xy isomorphic to Z, that R is normal, and that the quotient Gamma/R is isomorphic to Z/2Z. <br />
<br />
7. Suppose a group G has the property that every element in g satisfies g^2 = 1. Show that G must be abelian.<br />
<br />
==HOMEWORK 3 (due Oct 4)==<br />
<br />
1. Multiplication by -1 is an automorphism of Z/nZ. Let a be the homomorphism Z/2Z -> Aut(Z/nZ) which sends the nontrivial element of Z/2Z to multiplication by -1. Then we have a semidirect product of Z/pZ by Z/2Z as defined in class. This group of order 2n is called a ''dihedral group.'' and is denoted D_n (or sometimes D_{2n}).<br />
<br />
1a. Compute the center of D_n. (Note that the answer depends on n!)<br />
1b. Show that all involutions of D_n are conjugate to each other if and only if n is odd.<br />
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2. (More on direct products that we didn't do in class.) Suppose that G is a semidirect product of N with H. Suppose furthermore that H is also normal in G. Prove that G is a direct product of N with H.<br />
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3. Let Q be the quaternion group from last week's homework. Show that Q does not decompose as a semidirect product N \rtimes H, apart form the trivial cases N = {e} and N = Q.<br />
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4. The ''affine linear group'' of degree n is the group of transformations from R^n to R^n of the form x -> Ax + b, where A is a matrix in GL_n(R) and b is a vector in R^n. Prove that the affine linear group is a semidirect product R^n \rtimes GL_n.<br />
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5. The ''ordinary triangle group'' T(p,q,r) is the group with presentation <x,y | x^p = y^q = (xy)^r = 1>. This is a very interesting family of groups which arises naturally in many contexts in number theory and topology.<br />
<br />
6a. Prove that T(2,2,n) is isomorphic to the dihedral group D_n.<br />
6b. Prove that T(2,3,3) is finite and compute its order. Is it isomorphic to a finite group we've discussed in class?<br />
<br />
(It is a beautiful fact, but outside the scope of this course, that T(p,q,r) is finite if and only if 1/p + 1/q + 1/r > 1.)<br />
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7. Let G be a subgroup of GL_n(R) which contains SL_n(R). Prove that G is a normal subgroup of GL_n(R).<br />
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8. Let G be a group and let H_1 and H_2 be subgroups of G of finite index (that is, the number of cosets in G/H_i is finite, i = 1,2.) Prove that H_1 intersect H_2 also has finite index in G.<br />
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9. Recall that the commutator [x,y] is x y x^{-1} y^{-1}. If G is a group, the commutator subgroup of G, denoted G' or [G,G], is the subgroup of G generated by all commutators [x,y] for x,y in G.<br />
<br />
9a. Show that G' is a normal subgroup of G.<br />
9b. Show that G/G' is an abelian group.<br />
9c. Show that if f: G -> A is a homomorphism, with A abelian, then G' is contained in ker f. Conclude that f factors through a homomorphism G/G' -> A.<br />
9d. Show that G' = G if and only if G has no nontrivial abelian quotient. In this case, we say G is ''perfect''.<br />
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10. Under what conditions on (p,q,r) is the triangle group T(p,q,r) perfect?<br />
<br />
==HOMEWORK 4 (due Oct 16)==<br />
<br />
1. Let X be a set with a transitive action of a group G, which we think of as a homomorphism f: G -> Sym(X). Let H_x be the stabilizer of an element x of X.<br />
<br />
1a. If x' is another element of X, show that H_{x"} and H_x are conjugate subgroups. (Hint: this is not true without the hypothesis of transitivity, so make sure to use it!)<br />
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1b. Show that the kernel of f is the intersection of all the conjugates of H_x in G. (This subgroup is called the ''normal core'' of H_x.)<br />
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2. This week, a certified mathematical grown-up asked on MathOverflow what the index-3 subgroups of SL_2(Z/pZ) were. Let's do it ourselves for homework. Let p be a prime, and suppose H is an index-3 subgroup of SL_2(Z/pZ). I do not claim that the proof sketched below is necessarily the easiest, but it allows us to cover some ground that we won't get to do in lecture.<br />
<br />
2a. Using the coset action, show that the normal core of H is the kernel of a homomorphism f from SL_2(Z/pZ) to S_3.<br />
<br />
2b. A ''unipotent matrix'' is a matrix A such that (A-I)^n = 0 for some positive integer n. Prove that a unipotent element of SL_2(Z/pZ) has order dividing p.<br />
<br />
2c. You may use the fact that unipotent matrices generate SL_2(Z/pZ). Given this fact, show that f must be trivial when p > 3.<br />
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2d. Using the above, show that SL_2(Z/pZ) has no index-3 subgroups when p > 3.<br />
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2e. (extra credit) What can you say about the minimal index of a proper subgroup of SL_2(Z/pZ)?<br />
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3. Let H be a subgroup of G of index 2. Prove that H is normal.<br />
<br />
4. We proved two theorems in class: Lagrange's theorem, which says that the order of a subgroup of G divides |G|, and Cauchy's theorem, which says that if p divides |G| then there exists a subgroup of G of order p. Cauchy's theorem is a kind of partial converse to Lagrange's theorem, but the full converse is false. Give an example of a finite group G and a divisor n of |G| such that you can prove there is no subgroup of G of order n.<br />
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5. Let X be the set of lines in F_3^2 (here a line means a 1-dimensional linear subspace) so that |X| = 4. Let G be the projective linear group PGL_2(F_3), so that G acts on X via its action on F_3^2. This action can be thought of as a homomorphism from G to S_4. Show that this homomorphism is an ISOMORPHISM from PGL_2(F_3) to S_4.<br />
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6. The group S_3 has only three conjugacy classes. Prove that a finite group with at most three conjugacy classes has order at most 6.<br />
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7. Let X be the set of ordered triples of elements of {1,..,n}, for some n >= 3. Then S_n acts on X. How many orbits are there? (Hint: the answer does not depend on n.)<br />
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8. Suppose that G and H are groups and f: G -> H is a homomorphism. Recall the definition of the abelianization G^ab from the previous problem set.<br />
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8a. Show that the composition G -> H -> H^ab factors through a unique homomorphism G^ab -> H^ab, which we denote f^ab.<br />
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8b. Show that if f: G -> H and g: H -> Q are homomorphisms, then f^ab o g^ab = (f o g)^ab.<br />
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(For those reading MacLane, this constitutes a proof that abelianization is a '''functor''' from the category of groups to the category of abelian groups.)<br />
<br />
==HOMEWORK 5 (due Oct 23)==<br />
<br />
1. Show that there are only two different isomorphism classes of groups of order 14. (Hint: prove that such a group must be a semidirect product of a group of order 7 by a group of order 2.)<br />
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2. (variant of Burnside's Lemma) Let G be a finite group and H a normal subgroup. Suppose G acts on a finite set X. <br />
<br />
2a. Show that if x and y lie in the same H-orbit, and g is an element of G, then gx and gy also lie in the same H-orbit. Thus, G/H has a natural action on the set of H-orbits X/H.<br />
<br />
2b. Show that the average number of fixed points in X of an element of the coset gH is the same as the number of fixed points of the element gH of G/H in its action on X/H.<br />
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3. A ''central extension'' of a group G by an abelian group A is a group E, together with a surjective homomorphism E -> G whose kernel is central in E and is isomorphic to A.<br />
<br />
3a. Show that the quaternion group is a central extension of (Z/2Z)^2 by Z/2Z.<br />
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3b. Give an example of another nonabelian group which is a central extension of (Z/2Z)^2 by Z/2Z and which is not isomorphic to the quaternion group.<br />
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4. Prove that a central extension of an abelian group is nilpotent.<br />
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5. Give two different composition series for S_4 and show that they have the same composition factors.<br />
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6. Suppose that A is a finitely generated abelian group, and B is a subgroup of A.<br />
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6a. Prove that B is finitely generated and that rank(B) <= rank(A).<br />
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6b. Prove that rank(A/B) = rank(A) - rank(B).<br />
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7. For this problem, you may use the fact (which I should have asked on homework earlier) that if H_1 and H_2 are subgroups of G, then<br />
<br />
[G: H_1 intersect H_2] <= [G:H_1][G:H_2].<br />
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Prove that if G is a simple group, d is an integer greater than 1, and S is a subgroup of index d in G, then |G| <= d^d.<br />
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Conclude that, in particular, A_5 has no subgroup of index 3. On the other hand, exhibit a subgroup of A_5 of index 5.<br />
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(Note: d^d is not a sharp bound; can you get a bound of d factorial instead?)<br />
<br />
<!--(This argument shows that a solvable subgroup of A_n has size at most about (n-1)!, but this is far from sharp; e.g. I found a citation to a 1967 paper of Dixon which shows that a solvable subgroup of A_n has size at most k^n where k is around 2.88.)--><br />
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<!--<br />
8. Give an example of an abelian group A with a subgroup B such that B is not a direct summand of A.<br />
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9. Let A be the subgroup of Z^2 generated by the vector (1,0). Let B be the subgroup of Z^2 generated by the vector (3,3). Show that A and B are isomorphic to each other, but Z^2 / A is not isomorphic to Z^2 / B.<br />
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==HOMEWORK 6 (due Oct 30)==<br />
<br />
1. In this problem, you will construct a functor F from the category of finite sets to the category of complex vector spaces. I'll tell you what it does on objects: if X is a finite set, then F(X) is the vector space spanned by a set of basis elements {e_x}_{x in X}, so that dim F(X) = |X|. Your job: complete the definition of F by describing the map from Mor_{Finite Sets}(X,Y) to Mor_{Vector Spaces}(F(X),F(Y)), and showing that your map respects composition.<br />
<br />
2. If (V, rho) is a representation of a group G, denote by V^G the space of vectors in V such that gv = v for all g in G; these are called _invariant_ vectors. We saw that the permutation representation of S_3 has a one-dimensional space of invariants. Prove that if X is a finite set with G-action, and V_X the corresponding permutation representation of G, then<br />
<br />
dim V_X^G = number of orbits of X.<br />
<br />
3. (Invariants are functorial) Let G be a group. Then there is a category called C[G]-Mod whose objects are complex representations of G (i.e. pairs (V,rho) where V is a complex vector space and rho is a homomorphism from G to GL(V)). The morphisms in this category are the ones described in class: Mor((V,rho),(W,psi)) is the set of linear maps f from V to W such that<br />
<br />
f(rho(g)(v)) = psi(g)(f(v))<br />
<br />
for all g in G and all v in V.<br />
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Show that there is a functor H_0 from the category of representations of G to the category of vector spaces whose action on objects is given by<br />
<br />
H_0((V) = V^G.<br />
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(In other words: your job again is to explain how to associate to a morphism from (V,rho) to (W,phi) a morphism from V^G to W^G, in a way that's compatible with composition.)<br />
<br />
<!--4. The group PSL_2(Z) has a presentation of the form [U,V|U^2 = V^3 = 1]. Thus, an n-dimensional representation of PSL_2(Z) is precisely a pair of n x n matrices (rho(U),rho(V)), such that rho(U)^2 = rho(V)^3 = 1. Prove that for every n > 1 there are uncountably many isomorphism classes of n-dimensional representations of PSL_2(Z). (Extra credit: show that there are uncountably many isomorphism classes of _irreducible_ representations.)<br />
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An amazing theorem of Margulis (superrigidity) shows that SL_2(Z) is very special among arithmetic groups in this way. For instance, there are only _finitely_ many isomorphism classes of n-dimensional representations of PSL_d(Z) for any d > 2; those groups are "rigid."--><br />
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<!--<br />
5. Let G be a finite group, let X be a set with G-action, and let chi be the character of the corresponding permutation representation. Show that, for each g in G, chi(g) is the number of elements of X fixed by g. (I am actually going to prove this in class.)<br />
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6. Let V be the space of homogeneous degree-3 polynomials in variables x_1, x_2, x_3; then S_3 acts on V by permutation of the three variables. Describe the decomposition of V into irreducible representations of S_3.<br />
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7a. Show that if chi is the character of a representation of a finite group G, then chi(g^{-1}) is the complex conjugate of chi.<br />
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7b. As a consequence, show that if g is conjugate to g^{-1} for all g in G, then chi(g) is real.<br />
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7c. The groups D_p and S_n have the property that g is conjugate to g^{-1} for _every_ g in G, so that every character is a real-valued function on G. Give an example of a group G where this is not the case, and give an example of a representation of G whose character takes non-real values.<br />
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==HOMEWORK 7 (due Nov 6)==<br />
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1. Let G be a finite group, let X be a finite set with G-action, and let chi be the character of the corresponding permutation representation. Show that, for each g in G, chi(g) is the number of elements of X fixed by g. (I am actually going to prove this in class.)<br />
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2a. Show that if chi is the character of a finite-dimensional representation of a finite group G, then chi(g^{-1}) is the complex conjugate of chi(g) for all g.<br />
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2b. As a consequence, show that if g is conjugate to g^{-1} for all g in G, then chi(g) is real.<br />
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2c. The groups D_p and S_n have the property that g is conjugate to g^{-1} for _every_ g in G, so that every character is a real-valued function on G. Give an example of a group G where this is not the case, and give an example of a representation of G whose character takes non-real values.<br />
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3. Let V and W be complex vector spaces. Let f be a linear transformation of V, and g a linear transformation of W. <br />
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3a. Show that there is a unique linear transformation F satisfying<br />
<br />
F(v tensor w) = f(v) tensor g(w)<br />
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for all v in V and all w in W. We denote this transformation by f tensor g.<br />
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3b. Suppose V and W are finite dimensional. Show that Trace(f tensor g) = Trace(f) Trace(g). What does this say when f and g are both the identity transformation?<br />
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4. Let V be a vector space. We define Sym^2 V to be the quotient of V tensor V by the subspace generated by all elements of the form<br />
<br />
v tensor w - w tensor v<br />
<br />
for v,w in V.<br />
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Suppose dim V = n. What is dim Sym^2 V?<br />
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5. Show that Z/pZ tensor Z/qZ is zero when p and q are distinct primes.<br />
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==HOMEWORK 8 (due Nov 13)==<br />
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1. (Schur's lemma can be false over non-algebraically closed fields) Let G be Z/3Z and let g be a generator of G. Then G acts by cyclic permutations on the set {1,2,3}; let P be the corresponding permutation representation on the REAL (not complex!) vector space generated by e_1,e_2,e_3.<br />
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1a. Show that P breaks up as the direct sum of the trivial representation and a 2-dimensional representation V which is irreducible. (Remember, we are not working over the complex numbers, so you cannot use the criterion for irreducibility in terms of the character that we worked out in class for the complex case; you need to prove directly that V has no nontrivial proper subrepresentation.)<br />
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1b. Show that the map from V to V induced by g is G-equivariant, but is neither zero nor a scalar multiplication.<br />
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2. If V_1 and V_2 are representations (not necessarily irreducible!) of G, show that <chi_V_1, chi V_2> is a non-negative integer.<br />
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3. When a formula always yields a non-negative integer, it is often because it is secretly a formula for either the cardinality of some finite set or the dimension of some finite-dimensional vector space. In this case, show that<br />
<br />
<chi_{V_1}, chi_{V_2}> = dim_C Hom_G(V_1,V_2)<br />
<br />
where we recall that Hom_G(V_1,G_2) refers to the space of G-equivariant homomorphisms from V_1 to V_2.<br />
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4. If A is an abelian group, show that every irreducible representation of A is 1-dimensional. (Hint: show that there is a vector which is an eigenvector for every element of A!)<br />
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5. Let G be the group of order 20 which is the semidirect product of A by H, where A is Z/5Z and H is (Z/5Z)^* with its natural action on A. We will work out all the irreducible representations of A. <br />
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5a. First of all, consider the 5-dimensional permutation representation of G acting by left multiplication on the cosets of H. Show that this representation decomposes as a direct sum of the trivial representation and an irreducible 4-dimensional representation. (Hint: it is probably easier to use the character of this representation than to prove irreducibility directly.) <br />
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5b. Now show that there are 4 1-dimensional representations of G which factor through the quotient G/A (i.e. such that rho(A) is the identity.)<br />
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5c. Using the decomposition of the regular representation, show that the irreducible representations of G you have constructed are the only ones.<br />
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==HOMEWORK 9 (due Nov 20)==<br />
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0. Find all ring homomorphisms from Z to Z/30Z. [Warning: there are 8 of them, not 30.]<br />
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1. If R is a ring (not necessarily with 1) such that a^2 = a for every a in R, R is called a Boolean ring.<br />
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1a. Show that any nonzero Boolean ring has characteristic 2. (In other words, show that a+a=0 for all a in R.)<br />
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1b. Show that every Boolean ring is commutative.<br />
<br />
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2. Let phi : R -> S be a homomorphism of commutative rings with 1.<br />
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2a. If I is an ideal of S, show that phi^(-1) (I) is an ideal of R.<br />
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2b. If P is a prime ideal of S, show that phi^(-1) (P) is either R or a prime ideal of R.<br />
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2c. If M is a maximal ideal of S and phi is surjective, show that phi^(-1) (M) is a maximal ideal of R.<br />
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2d. Find a homomorphism phi : R -> S and a maximal ideal M of S such that phi^(-1) (M) is not a maximal ideal of R.<br />
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3. Let Nil(A) denote the set of nilpotent elements of a ring A, and let R be a commutative ring. In class it was observed that Nil(R) is an ideal of R.<br />
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3a. Find (a generator of) Nil(Z/720Z).<br />
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3b. Show that Nil(R/Nil(R)) = 0.<br />
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3c. If x is in Nil(R) and u is a unit, show that u+x is a unit.<br />
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3d. Show that Nil(S), where S is the 2x2 matrices with real coefficients, is not an ideal of S.<br />
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4a. If M is an R-module and I is an ideal of R such that im = 0 for all i in I and m in M, show that M carries the structure of an R/I-module.<br />
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4b. If S is a commutative ring with 1 and J is a maximal ideal of S, show that J/J^2 is a vector space over S/J.<br />
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5. A "divisible element" x of a Z-module is one such that for each positive integer k, there exists a y such that ky=x. Show that a free Z-module cannot contain a nonzero divisible element.<br />
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6. Prove that the direct sum of any collection of free R-modules is also a free R-module.<br />
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7. In this problem you will prove that the module M given by the direct product of copies of Z, indexed by the positive integers, is not a free Z-module. Therefore, assume that M is a free Z-module with basis B, and let N be the direct sum of copies of Z, as a submodule of M.<br />
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7a. Show there is a countable subset B1 of B such that N is contained in the submodule N1 generated by B1. [Hint: N is countable.]<br />
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7b. Show that M/N1 is a free Z-module.<br />
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7c. Let S = {(b1, b2, b3, ...) : b_i = i! or -i!}. Show that there is some s in S which is not in N1. [Hint: S is uncountable.]<br />
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7d. For any s in S but not N1, show that for each positive integer k, there exists some m in M with s-km in N1. Conclude that s+N1 is a divisible element of M/N1.<br />
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7e. Use problem 5 to obtain a contradiction.<br />
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8. In class I said "rank" is not always well-defined for R-modules if R is not commutative. Here is an example demonstrating this assertion. Let M be the countably infinite direct product of copies of Z as in exercise 7, and let R be the endomorphism ring of M. Define phi_1(a1,a2,a3,a4,...) = (a1,a3,a5,...) and phi_2(a1,a2,a3,a4,...) = (a2,a4,a6,...).<br />
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8a. Show that {phi_1, phi_2} is a free basis for R, as a left R-module. [Hint: let psi_1(a1,a2,a3,...) = (a1,0,a2,0,...) and psi_2(a1,a2,a3,...) = (0,a1,0,a2,...). Compute the products of the phi_i and psi_j in either order, and use the result to show that the phi_i are a basis.]<br />
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8b. Show that R is isomorphic to R^n for every positive integer n.<br />
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==HOMEWORK 10 (due Dec 4)==<br />
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1. '''How algebraists do calculus'''. The ''ring of dual numbers'' is the ring R = C[e]/(e^2). We think of e as an infinitesimal number; that is, something so small that its square is 0. This algebraic notion of infinitesimal is the basis for the algebraic notion of differentiation, which works as follows. Let f be a polynomial in R[x]. Show that there is a unique polynomial g in C[x] such that<br />
<br />
f(x+e) - f(x) = eg(x)<br />
<br />
and that in fact g is the derivative of f. In other words, the usual definition of "derivative" works just fine in this context, without any use of the notion of limit!<br />
<br />
2. Write down all the ideals of C[e]/e^2. (There are three.)<br />
<br />
3. Show that any right ideal of the ring M_2(Q) of 2x2 matrices is an ideal of the form I_V described in class for some subspace V of Q^2; that is, it consists precisely of those linear transformations whose image is contained in V.<br />
<br />
<br />
4. An ''idempotent'' element of a ring is an element e satisfying e^2 = e.<br />
<br />
4a. If e is an idempotent, show that 1-e is also an idempotent.<br />
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4b. Suppose R has a unit (we are actually always supposing this, but I just want to remind you of it for this problem) and suppose e is a ''central'' idempotent, i.e. an idempotent which commutes with all elements of R. Show that eR and (1-e)R are both two-sided ideals. Moreover, show that eR and (1-e)R are also subrings of R (with the warning that the identity in eR is not 1, which is not contained in eR, but rather e.)<br />
<br />
4c. Show that the map f_e: R -> eR defined by f_e(r) = er is a ring homomorphism. Finally show that the ring homomorphism<br />
<br />
f_{e} x f_{1-e}: R -> eR x (1-e)R <br />
<br />
is actually an isomorphism of rings. In other words, the presence of a central idempotent induces a decomposition a ring as a direct product.<br />
<br />
<br />
5. The ''center'' of a ring R is the subring of elements commuting with every element of R.<br />
<br />
5a. Show that the center of M_n(Q) is the ring of scalar matrices.<br />
<br />
5b. Show that the center of the group ring C[G] is the set of elements sum f(g) g such that f(g) is a class function.<br />
<br />
<br />
6. '''How algebraists do differential equations.''' We define the ''Weyl algebra'' W of differential operators in one variable as follows. Multiplication by x is a linear transformation on the complex vector space V of all smooth functions on C. Differentiation, which we denote d, is also a linear transformation on V. So we let W be the subring of End(V) generated by x and d. We note that x and d satisfy the relation xd = dx - 1; in particular, the Weyl algebra is not commutative.<br />
<br />
6a. Describe the submodule of V which is annihilated by the element d-x in W.<br />
<br />
6b. Show that, for any polynomial f(x), fd-df = -f', and also show that for any polynomial g(d), gx-xg = -g'. (This is another way algebraists do calculus!) <br />
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6c. Use the facts from (b) to show that the center of the Weyl algebra is just the field of constants. (Remark: you will use in the course of this proof the fact that a polynomial whose derivative is 0 is constant -- this would not be true if we were working in positive characteristic, as the example of x^p considered as a polynomial over F_p demonstrates.)<br />
<br />
<br />
7. If M is a left module for a ring R, we denote by Ann(M) the set of elements r in R such that rm = 0 for all m in M. Show that Ann(M) is a two-sided ideal. <br />
<br />
<br />
8. Let M be a left module for R, and let m be an element of M. We denote by Ann(m) the set of elements r in R such that rm = 0.<br />
<br />
8a. Show that Ann(m) is a left ideal, and show that the submodule Rm of M generated by m is isomorphic to the left module R/Ann(m).<br />
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8c. Let V,W be as in question 6. Show that Ann(e^x) is the left ideal W(d-1).<br />
<br />
==HOMEWORK 11 (due Dec 13)==<br />
<br />
1. Let f: R -> S be a homomorphism of rings. As discussed in class, if M is a left R-module, then S tensor_R M is a left S-module, and if N is an S-module, then N can also be considered as an R-module by pullback of the action.<br />
<br />
Prove that if M is a finitely generated left R-module, then S tensor_R M is a finitely generated left S-module. On the other hand, give an example to show that N can be finitely generated as a left S-module but not finitely generated when considered as a left R-module.<br />
<br />
2. Recall from class that when V and W are complex vector spaces, W^dual tensor V is naturally isomorphic to the space of linear homomorphisms Hom(W,V).<br />
<br />
Now suppose that V and W are left C[G]-modules, i.e. complex representations of a group G. Show that W^dual tensor_C[G} V is the maximal G-invariant quotient of Hom(W,V); in other words, the quotient of Hom(W,V) by the subspace generated by all functions F of the form F(w) = f(gw) - gf(w) for some function f in Hom(W,V) and some g in G.<br />
<br />
3. '''How algebraists do logic.''' Why are Boolean rings called Boolean rings? Because they model some features of the Boolean arithmetic that governs propositional logic. Let A be a collection of propositions. We define two operations on A as follows. By a + b we mean the negation of a xor b, while by ab we mean "a or b."<br />
<br />
3a. It is not a priori obvious that these operations satisfy the ring axioms, but assume this for the moment. Show that a^2 = a for all a (so A is a boolean ring) and 2a = 0 for all a (so A has characteristic 2.)<br />
<br />
3b. Suppose we have a consistent assignment of truth-values to the propositions in A. Show that the set of true statements of A forms an ideal of A.<br />
<br />
3c. (Optional for people who like propositional logic) Show that a(b+c) = ab + ac with the given operations.<br />
<br />
<br />
4. Let K be Q[x]/(x^3-2).<br />
<br />
4a. Show that K is a field.<br />
<br />
4b. Show that K tensor_Q K decomposes as the direct sum of two fields. What are they? [Note: K tensor_Q K is a ring!]<br />
<br />
5. There was some discussion in class about when we can "go back and forth" between right modules and left modules, as is the case with group rings.<br />
<br />
5a. If R is a ring, we denote R^op ("the opposite ring") to be the ring whose elements are those of R, but whose multiplication law x_opp is defined by s x_opp r = rs. Show that if M is a left R-module, then the map M x R^op -> M sending (x,r) to rx makes M into a right R^op module. In particular, if there is a RING ISOMORPHISM from R to R^op, then every left R-module can be thought of as a right R-module.<br />
<br />
5b. Verify that the map sending g to g^{-1} extends by linearity to give a ring isomorphism between C[G] and C[G]^op.<br />
<br />
5c. Construct a ring isomorphism between M_n(Q) and M_n(Q)^op.<br />
<br />
5d. Optional: Give an example of a ring (with 1) which is not isomorphic to its opposite ring. (This is not so easy, I think! I found a MathOverflow thread with several examples, none of which came with a short proof of the non-isomorphism -- try your hand!)<br />
<br />
6. . Let R be a commutative ring with 1. If R has a unique maximal ideal M, R is called a "local ring".<br />
<br />
6a. If R is a local ring, show that every element of R that is not in M is a unit.<br />
<br />
6b. Conversely, if R is such that the set of nonunits forms an ideal I, show that R is a local ring with maximal ideal I.<br />
<br />
7. The inclusion of S_3 into S_4 associated to the inclusion of {1,2,3} into {1,2,3,4} induces a ring homomorphism from C[S_3} to C{S_4]. Let V be a representation of C{S_3]. Then, as discussed in class, V tensor_C[S_3] C[S_4] is a representation of S_4, called the '''induced representation''' of V.<br />
<br />
7a. Show that dim_C (V tensor_C[S_3] C[S_4]) = 4 dim_C V.<br />
<br />
7b. When V is the trivial representation of S_3, decompose (V tensor_C[S_3] C[S_4]) as a sum of irreducible representations of S_4.<br />
<br />
--><br />
<br />
<!--==HOMEWORK PROBLEMS TO GIVE IN FUTURE SEMESTERS==<br />
<br />
Let f: C[x] tensor_C C[y] -> C[z] be the map of rings with f(x) = f(y) = z. Show that f(m tensor n) is not 0 for any pure tensor m tensor n, but f is not injective. (This is a warning not to try to check injectivity by checking that no pure tensors are killed by the map.)<br />
<br />
--></div>Dummithttps://hilbert.math.wisc.edu/wiki/index.php?title=741&diff=62117412013-11-01T15:51:20Z<p>Dummit: /* HOMEWORK 8 (due Nov 6) */</p>
<hr />
<div>'''Math 741'''<br />
<br />
Algebra<br />
<br />
Prof: [http://www.math.wisc.edu/~ellenber Jordan Ellenberg]<br />
<br />
Grader: Evan Dummit.<br />
* Homework policies:<br />
* Late homework may be given directly to the grader, along with either <br />
(i) the instructor's permission, or (ii) a polite request for mercy.<br />
* Assignments that are more than 1 page should be affixed in some reasonable way.<br />
* Results from places (e.g., the internet) other than 741 and standard books must be cited.<br />
<br />
JE's office hours: Monday 12pm-1pm (right after class)<br />
<br />
This course, the first semester of the introductory graduate sequence in algebra, will cover the basic theory of groups, group actions, representation, linear and multilinear algebra, and the beginnings of ring theory.<br />
<br />
==SYLLABUS==<br />
<br />
In this space we will record the theorems and definitions we covered each week, which we can use as a list of notions you should be prepared to answer questions about on the Algebra qualifying exam. The material covered on the homework is also an excellent guide to the scope of the course. <br />
<br />
'''WEEK 1''': <br />
<br />
Definition of group. Associativity. Inverse. <br />
<br />
Examples of group: GL_n(R). GL_n(Z). Z/nZ. R. Z. R^*. The free group F_k on k generators. <br />
<br />
Homomorphisms. The homomorphisms from F_k to G are in bijection with G^k. Isomorphisms.<br />
<br />
'''WEEK 2''':<br />
<br />
The symmetric group (or permutation group) S_n on n letters. Cycle decomposition of a permutation. Order of a permutation. Thm: every element of a finite group has finite order. <br />
<br />
Subgroups. Left and right cosets. Lagrange's Theorem. Cyclic groups. The order of an element of a finite group is a divisor of the order of the group.<br />
<br />
The sign homomorphism S_n -> +-1.<br />
<br />
<br />
'''WEEK 3'''<br />
<br />
Normal subgroups. The quotient of a group by a normal subgroup. The first isomorphism theorem. Examples of S_n -> +-1 and S_4 -> S_3 with kernel V_4, the Klein 4-group.<br />
<br />
Centralizers and centers. Abelian groups. The center of SL_n(R) is either 1 or +-1.<br />
<br />
Groups with presentations. The infinite dihedral group <x,y | x^2 = 1, y^2 = 1>.<br />
<br />
'''WEEK 4'''<br />
<br />
More on groups with presentations.<br />
<br />
Second and third isomorphism theorems.<br />
<br />
Semidirect products.<br />
<br />
'''WEEK 5'''<br />
<br />
Group actions, orbits, and stabilizers.<br />
<br />
Orbit-stabilizer theorem.<br />
<br />
Cayley's theorem.<br />
<br />
Cauchy's theorem.<br />
<br />
'''WEEK 6'''<br />
<br />
Applications of orbit-stabilizer theorem (p-groups have nontrivial center, first Sylow theorem.)<br />
<br />
Classification of finite abelian groups and finitely generated abelian groups.<br />
<br />
Composition series and the Jordan-Holder theorem (which we state but don't prove.) <br />
<br />
The difference between knowing the composition factors and knowing the group (e.g. all p-groups of the same order have the same composition factors.)<br />
<br />
'''WEEK 7'''<br />
<br />
Simplicity of A_n.<br />
<br />
Nilpotent groups (main example: the Heisenberg group)<br />
<br />
Derived series and lower central series.<br />
<br />
Category theory 101: Definition of category and functor. Some examples. A group is a groupoid with one object.<br />
<br />
'''WEEK 8'''<br />
<br />
Introduction to representation theory.<br />
<br />
'''WEEK 10'''<br />
<br />
Ring theory 101: Rings, ring homomorphisms, ideals, isomorphism theorems. Examples: fields, Z, the Hamilton quaternions, matrix rings, rings of polynomials and formal power series, quadratic integer rings, group rings. Integral domains. Maximal and prime ideals. The nilradical.<br />
<br />
Module theory 101: Modules, module homomorphisms, submodules, isomorphism theorems. Noetherian modules and Zorn's lemma. Direct sums and direct products of arbitrary collections of modules.<br />
<br />
<br />
==HOMEWORK 1 (due Sep 16)==<br />
<br />
1. Suppose that H_1 and H_2 are subgroups of a group G. Prove that the intersection of H_1 and H_2 is a subgroup of G.<br />
<br />
2. Recall that S_3 (the symmetric group) is the group of permutations of the set {1..3}. List all the subgroups of S_3.<br />
<br />
3. We can define an equivalence relation on rational numbers by declaring two rational numbers to be equal whenever they differ by an integer. We denote the set of equivalence classes by Q/Z. The operation of addition makes Q/Z into a group.<br />
<br />
a) For each n, prove that Q/Z has a subgroup of order n.<br />
<br />
b) Prove that Q/Z is a ''divisible'' group: that is, if x is an element of Q/Z and n is an integer, there exists an element y of Q/Z such that ny = x. (Note that we write the operation in this group as addition rather than multiplication, which is why we write ny for the n-fold product of y with itself rather than y^n)<br />
<br />
c) Prove that Q/Z is not finitely generated. (Hint: prove that if x_1, .. x_d is a finite subset of Q/Z, the subgroup of Q/Z generated by x_1, ... x_d is finite.)<br />
<br />
d) Conclude that Q is not finitely generated.<br />
<br />
4. We will prove that there is no homomorphism from SL_2(Z) to Z except the one which sends all of SL_2(Z) to 0. Suppose f is a homomorphism from SL_2(Z) to Z.<br />
<br />
a) Let U1 be the upper triangular matrix with 1's on the diagonal and a 1 in the upper right hand corner, as in class, and let U2 be the transpose of U1, also as in class. Show that (U1 U2^{-1})^6 = identity (JING 1, TAO 0) and explain why this implies that f(U1) = f(U2).<br />
<br />
b) Show that there is a matrix A in SL_2(Z) such that A U1 A^{-1} = U2^{-1}. (Recall that we say U1 and U2^{-1} are "conjugate".) Explain why this also implies that f(U1) = -f(U2).<br />
<br />
c) Explain why a) and b) imply that f must be identically 0.<br />
<br />
5. The argument above also shows that there is no nonzero homomorphism from SL_2(Z) to Z/pZ where p is a prime greater than 3. However, it leaves open the possibility that there is indeed a nonzero homomorphism from SL_2(Z) to Z/2Z. Exhibit such a homomorphism. Optional challenge problem: exhibit a nonzero homomorphism from SL_2(Z) to Z/3Z.<br />
<br />
<br />
==HOMEWORK 2 (due Sep 25)==<br />
<br />
1. Let V be the Klein 4-group in S_4. Let Q be the symmetric group on the set DF of double-flips in S_4; there are 3 double flips, so Q is isomorphic to S_3. If g is an element of S_4, we discussed in class that conjugation by g permutes the elements of DF. So to each element of g, we have associated an element f(g) of Q. More or less by definition, this defines a homomorphism f: G -> Q. Show that this homomorphism is surjective and has kernel H, and thus that G/H is isomorphic to Q.<br />
<br />
2. Let F_2 be the free group on two generators, which we denote x,y. Prove that there is a automorphism of F_2 which sends x to xyx and y to xy, and prove that this automorphism is unique.<br />
<br />
3. Let H be a subgroup of G, and let N_G(H), the ''normalizer'' of H in G, be the set of elements g in G satisfying g H g^{-1} = H. Prove that N_G(H) is a subgroup of G.<br />
<br />
4. Let T be the subgroup of GL_2(R) consisting of diagonal matrices. This is an example of a "Cartan subgroup," or "torus" (whence the notation T.) Describe the normalizer N(T) of T in GL_2(R). Prove that there is a homomorphism from N(T) to Z/2Z whose kernel is precisely T. <br />
<br />
5. Prove that the group of inner automorphisms of a group is normal in the full automorphism group Aut(G).<br />
<br />
6. Let H and H' be subgroups of a group G, and let x be an element of G. We denote by HxH' the set of elements of the form hxh', with h in H and h' in H', and we call HxH a ''double coset'' of the pair (H,H').<br />
<br />
a) Show that G decomposes as a disjoint union of double cosets of (H,H').<br />
<br />
b) Let G be GL_2(R) and let B be the subgroup of upper triangular matrices. Prove that G decomposes into exactly two double cosets of (B,B). (This is an example of the so-called *Bruhat decomposition* which is of great importance in the theory of algebraic groups and their representations.)<br />
<br />
c) Let G be the symmetric group S_n, and let H be the subgroup of permutations which fix 1. Describe the double coset decomposition of S_n into double cosets of (H,H).<br />
<br />
<br />
==HOMEWORK 3 (due Oct 2)==<br />
<br />
1. Compute the centralizer of the triple flip (12)(34)(56) in S_6. (Note: this is the kind of problem that a computer algebra package like sage is very good at, and I encourage you to learn to use sage well enough to check your answer.)<br />
<br />
2. Warning: a normal subgroup of a normal subgroup need not be normal! Let H be the subgroup of S_4 generated by (12)(34). Then H is a subgroup of the Klein 4-group V_4, which we have already shown is normal in S_4. Show that H is normal in V_4, but H is not normal in S_4.<br />
<br />
3. Let G be a group and let H_1 and H_2 be subgroups of G of finite index (that is, the number of cosets in G/H_i is finite, i = 1,2.) Prove that H_1 intersect H_2 also has finite index in G.<br />
<br />
4. Let X be the set of lines in F_3^2 (here a line means a 1-dimensional linear subspace) so that |X| = 4. Let G be the projective linear group PGL_2(F_3), so that G acts on X via its action on F_3^2. This action can be thought of as a homomorphism from G to S_4. Show that this homomorphism is an ISOMORPHISM from PGL_2(F_3) to S_4.<br />
<br />
5. We proved two theorems in class: Lagrange's theorem, which says that the order of a subgroup of G divides |G|, and Cauchy's theorem, which says that if p divides |G| then there exists a subgroup of G of order p. Cauchy's theorem is a kind of partial converse to Lagrange's theorem, but the full converse is false. Give an example of a finite group G and a divisor n of |G| such that you can prove there is no subgroup of G of order n.<br />
<br />
6. The group S_3 has only three conjugacy classes. Prove that a finite group with at most three conjugacy classes has order at most 6.<br />
<br />
7. Let X be the set of ordered triples of elements of {1,..,n}, for some n >= 3. Then S_n acts on X. How many orbits are there? (Hint: the answer does not depend on n.) OPTIONAL: How many orbits are there on the space of ordered k-tuples, when n >= k? <br />
<br />
==HOMEWORK 4 (due Oct 9)==<br />
<br />
1. Recall that the commutator [x,y] is x y x^{-1} y^{-1}. If G is a group, the commutator subgroup of G, denoted G' or [G,G], is the subgroup of G generated by all commutators [x,y] for x,y in G.<br />
<br />
1a. Show that G' is a normal subgroup of G.<br />
1b. Show that G/G' is an abelian group, which we call the abelianization, G^ab.<br />
1c. Show that if f: G -> A is a homomorphism, with A abelian, then G' is contained in ker f. Conclude that f factors through a homomorphism G/G' -> A.<br />
1d. Show that G' = G if and only if G has no nontrivial abelian quotient. In this case, we say G is ''perfect''.<br />
<br />
2. Prove that any p-Sylow subgroup of GL_n(F_p) is contained in SL_n(F_p).<br />
<br />
3. Let U_n(F_p) be the subgroup of GL_n(F_p) consisting of upper triangular matrices with 1's on the diagonal. Show that U_n(F_p) is a p-Sylow subgroup of GL_n(F_p).<br />
<br />
4. U_n(F_p) is called the *unipotent subgroup* of GL_n(F_p), because it has the property (not part of your homework, but easy to check) that every u in U_n(F_p) is unipotent, i.e. its characteristic polynomial is (x-1)^n. It is a p-group (as you know by virtue of the last problem) so it must have a nontrivial center. What is the center of U_nF_p)?<br />
<br />
5. U_3(F_p) is Shamgar Gurevich's favorite group, the *Heisenberg group" of order p^3. Write down a composition series for U_3(F_p).<br />
<br />
6. Let A be the subgroup of Z^2 generated by the vector (1,0). Let B be the subgroup of Z^2 generated by the vector (3,3). Show that A and B are isomorphic to each other, but Z^2 / A is not isomorphic to Z^2 / B.<br />
<br />
7. Suppose that G and H are groups and f: G -> H is a homomorphism. Recall the definition of the abelianization G^ab from problem 1.<br />
<br />
7a. Show that the composition G -> H -> H^ab factors through a unique homomorphism G^ab -> H^ab, which we denote f^ab.<br />
7b. Show that if f: G -> H and g: H -> Q are homomorphisms, then f^ab o g^ab = (f o g)^ab.<br />
<br />
(For those reading MacLane, this constitutes a proof that abelianization is a '''functor''' from the category of groups to the category of abelian groups.)<br />
<br />
8. Give two different composition series for S_4 and show that they have the same composition factors.<br />
<br />
OPTIONAL: We saw in class that a finitely generated abelian group that was torsion-free was in fact a free abelian group. This totally dies without the hypothesis of abelianness. To see this, give an example of a finitely generated group which is torsion free but which is not a free group, and which does not even *contain* any free group of rank greater than 1.<br />
<br />
==HOMEWORK 5 (due Oct 16)==<br />
<br />
1. Let G be a group, thought of as a category with one object. Show that a set with an action of G is the same thing as a functor from G to the category of sets.<br />
<br />
2. If (V, rho) is a representation of a group G, denote by V^G the space of vectors in V such that gv = v for all g in G; these are called _invariant_ vectors. We saw that the permutation representation of S_3 has a one-dimensional space of invariants. Prove that if X is a finite set with G-action, and V_X the corresponding permutation representation of G, then<br />
<br />
dim V_X^G = number of orbits of X.<br />
<br />
3. (Invariants are functorial) Let G be a group. Then there is a category called C[G]-Mod whose objects are complex representations of G (i.e. pairs (V,rho) where V is a complex vector space and rho is a homomorphism from G to GL(V)). The morphisms in this category are the ones described in class: Mor((V,rho),(W,psi)) is the set of linear maps f from V to W such that<br />
<br />
f(rho(g)(v)) = psi(g)(f(v))<br />
<br />
for all g in G and all v in V.<br />
<br />
Show that there is a functor H_0 from the category of representations of G to the category of vector spaces whose action on objects is given by<br />
<br />
H_0((V) = V^G.<br />
<br />
(In other words: your job again is to explain how to associate to a morphism from (V,rho) to (W,phi) a morphism from V^G to W^G, in a way that's compatible with composition.)<br />
<br />
4. Let G be a finite group, let X be a set with G-action, and let chi be the character of the corresponding permutation representation. Show that, for each g in G, chi(g) is the number of elements of X fixed by g. (I am actually going to prove this in class.)<br />
<br />
5. Let V be the space of homogeneous degree-3 polynomials in variables x_1, x_2, x_3; then S_3 acts on V by permutation of the three variables. Describe the decomposition of V into irreducible representations of S_3.<br />
<br />
6a. Show that if chi is the character of a representation of a finite group G, then chi(g^{-1}) is the complex conjugate of chi.<br />
<br />
6b. As a consequence, show that if g is conjugate to g^{-1} for all g in G, then chi(g) is real.<br />
<br />
6c. The groups D_p and S_n have the property that g is conjugate to g^{-1} for _every_ g in G, so that every character is a real-valued function on G. Give an example of a group G where this is not the case, and give an example of a representation of G whose character takes non-real values.<br />
<br />
OPTIONAL: (for people who know some topology.) Let $X$ be a topological space and let $Paths/X$ be the category of paths up to isotopy as described in class. Explain how to construct a bijection between {functors from Paths/X to the category FinSet of finite sets} and {finite covering spaces of X}. (For simplicity you may assume X is path-connected.)<br />
<br />
==HOMEWORK 6 (due Oct 23)==<br />
<br />
1. Let V_k be the space of homogenous degree-k polynomials in x_1, x_2, x_3. Compute the character of S_3 acting on V_k, and, using this, give formulas for the multiplicity of the trivial rep, the sign rep, and the standard rep of S_3 in V_k.<br />
<br />
2. (Schur's lemma can be false over non-algebraically closed fields) Let G be Z/3Z and let g be a generator of G. Then G acts by cyclic permutations on the set {1,2,3}; let P be the corresponding permutation representation on the REAL (not complex!) vector space generated by e_1,e_2,e_3.<br />
<br />
2a. Show that P breaks up as the direct sum of the trivial representation and a 2-dimensional representation V which is irreducible. (Remember, we are not working over the complex numbers, so you cannot use the criterion for irreducibility in terms of the character that we worked out in class for the complex case; you need to prove directly that V has no nontrivial proper subrepresentation.)<br />
<br />
2b. Show that the map from V to V induced by g is G-equivariant, but is neither zero nor a scalar multiplication.<br />
<br />
<br />
The following two problems involve the tensor product. I am going to teach the general notion of tensor product of modules in this course, but I am expecting you already know what the tensor product of vector spaces. If not, please review it by learning the definition and doing these problems, because having a familiarity with the vector space theory will be very helpful in learning the general story.<br />
<br />
3. Let V and W be complex vector spaces. Let f be a linear transformation of V, and g a linear transformation of W. <br />
<br />
3a. Show that there is a unique linear transformation F satisfying<br />
<br />
F(v tensor w) = f(v) tensor g(w)<br />
<br />
for all v in V and all w in W. We denote this transformation by f tensor g.<br />
<br />
3b. Suppose V and W are finite dimensional. Show that Trace(f tensor g) = Trace(f) Trace(g). What does this say when f and g are both the identity transformation?<br />
<br />
4. Let V be a vector space. We define Sym^2 V to be the quotient of V tensor V by the subspace generated by all elements of the form<br />
<br />
v tensor w - w tensor v<br />
<br />
for v,w in V.<br />
<br />
Suppose dim V = n. What is dim Sym^2 V?<br />
<br />
5. For any n, the group S_n has an (n-1)-dimensional representation, the "standard representation," on the space of vectors in C^n whose coordinates sum to 0. When n=3, we proved in class that this representation was irreducible. Prove that the standard representation is irreducible for all n.<br />
<br />
6. Give an example showing that the standard representation of S_n need NOT be irreducible over a field of characteristic p.<br />
<br />
7. Let H be a subgroup of G and let V be a representation of G. Let chi_{G/H} be the character of the permutation representation of G on the set of cosets G/H, so that, by problem 1, chi_{G/H}(g) is the number of fixed points of g in its action on the cosets. As discussed in class, the dimension of the G-invariant subspace V^G is given by the inner product of chi_V with the trivial character. Prove that the dimension of V^H is given by the inner product of chi_V with chi_{G/H}.<br />
<br />
==HOMEWORK 7 (due Oct 30)==<br />
<br />
1. When a formula always yields a non-negative integer, it is often because it is secretly a formula for either the cardinality of some finite set or the dimension of some finite-dimensional vector space. In this case, show that<br />
<br />
<chi_{V_1}, chi_{V_2}> = dim_C Hom_G(V_1,V_2)<br />
<br />
where we recall that Hom_G(V_1,G_2) refers to the space of G-equivariant homomorphisms from V_1 to V_2.<br />
<br />
2. Let G be the group of affine linear transformations of F_5; that is, it is the group of transformations x -> ax + b where a lies in (Z/5Z)^* and b lies in (Z/5Z). Note that G has order 20. Write H for the subgroup of G consisting of transformations fixing 0 (i.e. those for which b=0). <br />
<br />
We will work out all the irreducible representations of G. <br />
<br />
2a. First of all, consider the 5-dimensional permutation representation of G acting by left multiplication on the cosets of H. Show that this representation decomposes as a direct sum of the trivial representation and an irreducible 4-dimensional representation. (Hint: it is probably easier to use the character of this representation than to prove irreducibility directly.) <br />
<br />
2b. Now show that there are 4 1-dimensional representations of G.<br />
<br />
2c. Show that the irreducible representations of G you have constructed are the only ones.<br />
<br />
3. If G and H are finite groups, and V is a representation of G, and W is a representation of H, then V tensor W is a representation of GxH, and the character chi_{V tensor W} is given by<br />
<br />
chi_{V tensor W}(g,h) = chi_V(g) chi_W(h).<br />
<br />
3a. Prove that V tensor W is irreducible if and only if V and W are irreducible.<br />
<br />
3b. Prove that all the irreducible representations of G x H are of the form V tensor W.<br />
<br />
(Together, these propositions show that you can completely describe the representation theory of G x H in terms of that of G and that of H.)<br />
<br />
4. Suppose that V is an irreducible representation of S_n, and suppose that when we consider V as a representation of the alternating group A_n it is NOT irreducible. Prove that V, considered as a representation of A_n, is the direct sum of two non-isomorphic irreducible representations. Prove furthermore that chi_V(g) = 0 for all odd permutations g. Give an example of such an irreducible representation of S_4.<br />
<br />
<br />
OPTIONAL (because I couldn't quickly think of an easy way to do it!) Prove that there exists such a representation of S_n for every n >= 3.<br />
<br />
5. (Fourier analysis over finite fields.) I said in class that the representation theory that goes into Fourier analysis is different from the representation theory of finite groups we do in class, but that's not quite true; when you do Fourier theory over finite fields, the two theories come into much closer contact, with the bonus that we don't have to worry about issues of infinite sums that I hand-waved away in class.<br />
<br />
Let F be the field of p elements and let V be the space of complex-valued functions on F; so V is a p-dimensional space. Let G be the group Z/pZ (in other words, it is the additive group of the field.)<br />
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Now V is a representation of G, in the same way we discussed in class: if f is a function in V, and a is an element of g, then the function gf is defined by<br />
<br />
gf(x) = f(x+a)<br />
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5a. Describe the breakup of V into irreducible representations of G, which are all 1-dimensional (not just what the characters are, explicitly decribe the irreducible constituents of V as subspaces of V!)<br />
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5b. There is a natural norm on V which sends f to ||f|| = sum_x |f(x)|^2.<br />
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Given any f, we have a unique decomposition f = sum_i f_i, where f_i lies in the irreducible constituent V_i of V.<br />
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Prove that ||f|| = sum_i ||f_i||.<br />
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OPTIONAL (for people who are taking analysis) Explain why ||f_i|| is correctly thought of as a "Fourier coefficient" and why the fact proved in 5b is the finite-field analogue of Parseval's identity.<br />
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==HOMEWORK 8 (due Nov 6)==<br />
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1. If R is a ring (not necessarily with 1) such that a^2 = a for every a in R, R is called a Boolean ring.<br />
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1a. Show that any nonzero Boolean ring has characteristic 2. (In other words, show that a+a=0 for all a in R.)<br />
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1b. Show that every Boolean ring is commutative.<br />
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2. Let phi : R -> S be a homomorphism of commutative rings (recalling that rings for us have a multiplicative identity 1 and that ring homomorphisms take 1 to 1.)<br />
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2a. If I is an ideal of S, show that phi^(-1) (I) is an ideal of R.<br />
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2b. If P is a prime ideal of S, show that phi^(-1) (P) is either R or a prime ideal of R.<br />
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2c. If M is a maximal ideal of S and phi is surjective, show that phi^(-1) (M) is a maximal ideal of R.<br />
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2d. Find a homomorphism phi : R -> S and a maximal ideal M of S such that phi^(-1) (M) is not a maximal ideal of R.<br />
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3. Let Nil(R) denote the set of nilpotent elements of a ring R; that is, the set of x such that x^k = 0 for some positive integer k. <br />
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3a. Show that Nil(R) is an ideal whenever R is a commutative ring.<br />
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3b. Describe Nil(Z/720Z).<br />
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3c. If x is in Nil(R) and u is a unit, show that u+x is a unit.<br />
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3d. Show that Nil(M_2(R)) is not an ideal of the matrix algebra M_2(R). (That is, it is neither a left nor a right ideal.)<br />
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4. We say a ring is Noetherian if any ascending chain of ideals I_1 < I_2 < I_3 < …. eventually stabilizes. This is a somewhat funny-looking condition if you haven't encountered it before, but it turns out to be a very useful way of formalizing the notion that a ring is "not too bad."<br />
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4a. Show that Z is Noetherian.<br />
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4b. Let R be the ring of continuous functions from the real numbers to the real numbers. Show that R is not Noetherian.<br />
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5. '''How algebraists do calculus'''. The ''ring of dual numbers'' is the ring A = C[e]/(e^2). We think of e as an infinitesimal number; that is, something so small that its square is 0. This algebraic notion of infinitesimal is the basis for the algebraic notion of differentiation, which works as follows. Let f be a polynomial in C[x]. Show that there is a unique polynomial g in C[x] such that<br />
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f(x+e) - f(x) = eg(x)<br />
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and that in fact g is the derivative of f. In other words, the usual definition of "derivative" works just fine in this context, without any use of the notion of limit!<br />
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6. Write down all the ideals of C[e]/e^2. (There are three.)<br />
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7. Show that any right ideal of the ring M_2(Q) of 2x2 matrices is an ideal of the form I_V described in class for some subspace V of Q^2; that is, it consists precisely of those linear transformations whose image is contained in V.<br />
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<!--Below you will find a repository of homework problems. Note that some of these problems are taken from Lang's ''Algebra''.<br />
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==HOMEWORK 1 (due Sep 20)==<br />
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1. Suppose that H_1 and H_2 are subgroups of a group G. Prove that the intersection of H_1 and H_2 is a subgroup of G.<br />
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2. Recall that S_3 (the symmetric group) is the group of permutations of the set {1..3}. List all the subgroups of S_3.<br />
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3. We can define an equivalence relation on rational numbers by declaring two rational numbers to be equal whenever they differ by an integer. We denote the set of equivalence classes by Q/Z. The operation of addition makes Q/Z into a group.<br />
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a) For each n, prove that Q/Z has a subgroup of order n.<br />
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b) Prove that Q/Z is a ''divisible'' group: that is, if x is an element of Q/Z and n is an integer, there exists an element y of Q/Z such that ny = x. (Note that we write the operation in this group as addition rather than multiplication, which is why we write ny for the n-fold product of y with itself rather than y^n)<br />
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c) Prove that Q/Z is not finitely generated. (Hint: prove that if x_1, .. x_d is a finite subset of Q/Z, the subgroup of Q/Z generated by x_1, ... x_d is finite.)<br />
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4. Let F_2 be the free group on two generators x,y. Prove that there is a automorphism of F_2 which sends x to xyx and y to xy, and prove that this automorphism is unique.<br />
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5. Let H be a subgroup of G, and let N_G(H), the ''normalizer'' of H in G, be the set of elements g in G satsifying g H g^{-1} = H. Prove that N_G(H) is a subgroup of G.<br />
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6. Let T be the subgroup of GL_2(R) consisting of diagonal matrices. This is an example of a "Cartan subgroup," or "torus" (whence the notation T.) Describe the normalizer N(T) of T in GL_2(R). Prove that there is a homomorphism from N(T) to Z/2Z whose kernel is precisely T. <br />
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7. Prove that the group of inner automorphisms of a group is normal in the full automorphism group Aut(G).<br />
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8. Let H and H' be subgroups of a group G, and let x be an element of G. We denote by HxH' the set of elements of the form hxh', with h in H and h' in H', and we call HxH a ''double coset'' of the pair (H,H').<br />
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a) Show that G decomposes as a disjoint union of double cosets of (H,H').<br />
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b) Let G be GL_2(R) and let B be the subgroup of upper triangular matrices. Prove that G decomposes into exactly two double cosets of (B,B).<br />
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c) Let G be the symmetric group S_n, and let H be the subgroup of permutations which fix 1. Describe the double coset decomposition of S_n into double cosets of (H,H).<br />
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==HOMEWORK 2 (due Sep 27)==<br />
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1. Prove that the symmetric group S_n is generated by transpositions. Hint: proceed by induction on n. Let H be the subgroup of S_n consisting of permutations fixing 1. Observe that H is isomorphic to S_{n-1}, so that H is generated by transpositions. Explain why it suffices to show that, for any g in S_n, the coset gH contains a transposition. Then explain why this is the case.<br />
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2. Given that the symmetric group S_n is generated by transpositions, show that there are no nontrivial homomorphisms from S_n to Z/pZ for any odd prime p.<br />
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3. Compute the centralizer of the triple flip (12)(34)(56) in S_6. (Note: this is the kind of problem that a computer algebra package like sage is very good at, and I encourage you to learn to use sage well enough to check your answer.)<br />
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4. Warning: a normal subgroup of a normal subgroup need not be normal! Let H be the subgroup of S_4 generated by (12)(34). Then H is a subgroup of the Klein 4-group V_4, which we have already shown is normal in S_4. Show that H is normal in V_4, but H is not normal in S_4.<br />
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5. The '''quaternion group''' Q is a non-abelian finite group of order 8. Its elements are 1, -1, i, -i, j, -j, k, -k; it satisfies g^2 = -1 for all g except +-1, and ijk = 1. (Note that the notation "-i" means "the product of i with -1" and -1 is understood to be central in Q.) Prove that every subgroup of Q is normal.<br />
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6. (finishing example done in class) Let Gamma be the group F<x,y> / (x^2 = 1, y^2 = 1). (Remember, this means the quotient of the free group by the group generated by all conjugates of x^2 and y^2.) Prove that Gamma is infinite. Prove that Gamma has a subgroup R generated by xy isomorphic to Z, that R is normal, and that the quotient Gamma/R is isomorphic to Z/2Z. <br />
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7. Suppose a group G has the property that every element in g satisfies g^2 = 1. Show that G must be abelian.<br />
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==HOMEWORK 3 (due Oct 4)==<br />
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1. Multiplication by -1 is an automorphism of Z/nZ. Let a be the homomorphism Z/2Z -> Aut(Z/nZ) which sends the nontrivial element of Z/2Z to multiplication by -1. Then we have a semidirect product of Z/pZ by Z/2Z as defined in class. This group of order 2n is called a ''dihedral group.'' and is denoted D_n (or sometimes D_{2n}).<br />
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1a. Compute the center of D_n. (Note that the answer depends on n!)<br />
1b. Show that all involutions of D_n are conjugate to each other if and only if n is odd.<br />
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2. (More on direct products that we didn't do in class.) Suppose that G is a semidirect product of N with H. Suppose furthermore that H is also normal in G. Prove that G is a direct product of N with H.<br />
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3. Let Q be the quaternion group from last week's homework. Show that Q does not decompose as a semidirect product N \rtimes H, apart form the trivial cases N = {e} and N = Q.<br />
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4. The ''affine linear group'' of degree n is the group of transformations from R^n to R^n of the form x -> Ax + b, where A is a matrix in GL_n(R) and b is a vector in R^n. Prove that the affine linear group is a semidirect product R^n \rtimes GL_n.<br />
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5. The ''ordinary triangle group'' T(p,q,r) is the group with presentation <x,y | x^p = y^q = (xy)^r = 1>. This is a very interesting family of groups which arises naturally in many contexts in number theory and topology.<br />
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6a. Prove that T(2,2,n) is isomorphic to the dihedral group D_n.<br />
6b. Prove that T(2,3,3) is finite and compute its order. Is it isomorphic to a finite group we've discussed in class?<br />
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(It is a beautiful fact, but outside the scope of this course, that T(p,q,r) is finite if and only if 1/p + 1/q + 1/r > 1.)<br />
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7. Let G be a subgroup of GL_n(R) which contains SL_n(R). Prove that G is a normal subgroup of GL_n(R).<br />
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8. Let G be a group and let H_1 and H_2 be subgroups of G of finite index (that is, the number of cosets in G/H_i is finite, i = 1,2.) Prove that H_1 intersect H_2 also has finite index in G.<br />
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9. Recall that the commutator [x,y] is x y x^{-1} y^{-1}. If G is a group, the commutator subgroup of G, denoted G' or [G,G], is the subgroup of G generated by all commutators [x,y] for x,y in G.<br />
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9a. Show that G' is a normal subgroup of G.<br />
9b. Show that G/G' is an abelian group.<br />
9c. Show that if f: G -> A is a homomorphism, with A abelian, then G' is contained in ker f. Conclude that f factors through a homomorphism G/G' -> A.<br />
9d. Show that G' = G if and only if G has no nontrivial abelian quotient. In this case, we say G is ''perfect''.<br />
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10. Under what conditions on (p,q,r) is the triangle group T(p,q,r) perfect?<br />
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==HOMEWORK 4 (due Oct 16)==<br />
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1. Let X be a set with a transitive action of a group G, which we think of as a homomorphism f: G -> Sym(X). Let H_x be the stabilizer of an element x of X.<br />
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1a. If x' is another element of X, show that H_{x"} and H_x are conjugate subgroups. (Hint: this is not true without the hypothesis of transitivity, so make sure to use it!)<br />
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1b. Show that the kernel of f is the intersection of all the conjugates of H_x in G. (This subgroup is called the ''normal core'' of H_x.)<br />
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2. This week, a certified mathematical grown-up asked on MathOverflow what the index-3 subgroups of SL_2(Z/pZ) were. Let's do it ourselves for homework. Let p be a prime, and suppose H is an index-3 subgroup of SL_2(Z/pZ). I do not claim that the proof sketched below is necessarily the easiest, but it allows us to cover some ground that we won't get to do in lecture.<br />
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2a. Using the coset action, show that the normal core of H is the kernel of a homomorphism f from SL_2(Z/pZ) to S_3.<br />
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2b. A ''unipotent matrix'' is a matrix A such that (A-I)^n = 0 for some positive integer n. Prove that a unipotent element of SL_2(Z/pZ) has order dividing p.<br />
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2c. You may use the fact that unipotent matrices generate SL_2(Z/pZ). Given this fact, show that f must be trivial when p > 3.<br />
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2d. Using the above, show that SL_2(Z/pZ) has no index-3 subgroups when p > 3.<br />
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2e. (extra credit) What can you say about the minimal index of a proper subgroup of SL_2(Z/pZ)?<br />
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3. Let H be a subgroup of G of index 2. Prove that H is normal.<br />
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4. We proved two theorems in class: Lagrange's theorem, which says that the order of a subgroup of G divides |G|, and Cauchy's theorem, which says that if p divides |G| then there exists a subgroup of G of order p. Cauchy's theorem is a kind of partial converse to Lagrange's theorem, but the full converse is false. Give an example of a finite group G and a divisor n of |G| such that you can prove there is no subgroup of G of order n.<br />
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5. Let X be the set of lines in F_3^2 (here a line means a 1-dimensional linear subspace) so that |X| = 4. Let G be the projective linear group PGL_2(F_3), so that G acts on X via its action on F_3^2. This action can be thought of as a homomorphism from G to S_4. Show that this homomorphism is an ISOMORPHISM from PGL_2(F_3) to S_4.<br />
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6. The group S_3 has only three conjugacy classes. Prove that a finite group with at most three conjugacy classes has order at most 6.<br />
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7. Let X be the set of ordered triples of elements of {1,..,n}, for some n >= 3. Then S_n acts on X. How many orbits are there? (Hint: the answer does not depend on n.)<br />
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8. Suppose that G and H are groups and f: G -> H is a homomorphism. Recall the definition of the abelianization G^ab from the previous problem set.<br />
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8a. Show that the composition G -> H -> H^ab factors through a unique homomorphism G^ab -> H^ab, which we denote f^ab.<br />
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8b. Show that if f: G -> H and g: H -> Q are homomorphisms, then f^ab o g^ab = (f o g)^ab.<br />
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(For those reading MacLane, this constitutes a proof that abelianization is a '''functor''' from the category of groups to the category of abelian groups.)<br />
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==HOMEWORK 5 (due Oct 23)==<br />
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1. Show that there are only two different isomorphism classes of groups of order 14. (Hint: prove that such a group must be a semidirect product of a group of order 7 by a group of order 2.)<br />
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2. (variant of Burnside's Lemma) Let G be a finite group and H a normal subgroup. Suppose G acts on a finite set X. <br />
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2a. Show that if x and y lie in the same H-orbit, and g is an element of G, then gx and gy also lie in the same H-orbit. Thus, G/H has a natural action on the set of H-orbits X/H.<br />
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2b. Show that the average number of fixed points in X of an element of the coset gH is the same as the number of fixed points of the element gH of G/H in its action on X/H.<br />
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3. A ''central extension'' of a group G by an abelian group A is a group E, together with a surjective homomorphism E -> G whose kernel is central in E and is isomorphic to A.<br />
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3a. Show that the quaternion group is a central extension of (Z/2Z)^2 by Z/2Z.<br />
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3b. Give an example of another nonabelian group which is a central extension of (Z/2Z)^2 by Z/2Z and which is not isomorphic to the quaternion group.<br />
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4. Prove that a central extension of an abelian group is nilpotent.<br />
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5. Give two different composition series for S_4 and show that they have the same composition factors.<br />
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6. Suppose that A is a finitely generated abelian group, and B is a subgroup of A.<br />
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6a. Prove that B is finitely generated and that rank(B) <= rank(A).<br />
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6b. Prove that rank(A/B) = rank(A) - rank(B).<br />
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7. For this problem, you may use the fact (which I should have asked on homework earlier) that if H_1 and H_2 are subgroups of G, then<br />
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[G: H_1 intersect H_2] <= [G:H_1][G:H_2].<br />
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Prove that if G is a simple group, d is an integer greater than 1, and S is a subgroup of index d in G, then |G| <= d^d.<br />
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Conclude that, in particular, A_5 has no subgroup of index 3. On the other hand, exhibit a subgroup of A_5 of index 5.<br />
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(Note: d^d is not a sharp bound; can you get a bound of d factorial instead?)<br />
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<!--(This argument shows that a solvable subgroup of A_n has size at most about (n-1)!, but this is far from sharp; e.g. I found a citation to a 1967 paper of Dixon which shows that a solvable subgroup of A_n has size at most k^n where k is around 2.88.)--><br />
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<!--<br />
8. Give an example of an abelian group A with a subgroup B such that B is not a direct summand of A.<br />
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9. Let A be the subgroup of Z^2 generated by the vector (1,0). Let B be the subgroup of Z^2 generated by the vector (3,3). Show that A and B are isomorphic to each other, but Z^2 / A is not isomorphic to Z^2 / B.<br />
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==HOMEWORK 6 (due Oct 30)==<br />
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1. In this problem, you will construct a functor F from the category of finite sets to the category of complex vector spaces. I'll tell you what it does on objects: if X is a finite set, then F(X) is the vector space spanned by a set of basis elements {e_x}_{x in X}, so that dim F(X) = |X|. Your job: complete the definition of F by describing the map from Mor_{Finite Sets}(X,Y) to Mor_{Vector Spaces}(F(X),F(Y)), and showing that your map respects composition.<br />
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2. If (V, rho) is a representation of a group G, denote by V^G the space of vectors in V such that gv = v for all g in G; these are called _invariant_ vectors. We saw that the permutation representation of S_3 has a one-dimensional space of invariants. Prove that if X is a finite set with G-action, and V_X the corresponding permutation representation of G, then<br />
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dim V_X^G = number of orbits of X.<br />
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3. (Invariants are functorial) Let G be a group. Then there is a category called C[G]-Mod whose objects are complex representations of G (i.e. pairs (V,rho) where V is a complex vector space and rho is a homomorphism from G to GL(V)). The morphisms in this category are the ones described in class: Mor((V,rho),(W,psi)) is the set of linear maps f from V to W such that<br />
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f(rho(g)(v)) = psi(g)(f(v))<br />
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for all g in G and all v in V.<br />
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Show that there is a functor H_0 from the category of representations of G to the category of vector spaces whose action on objects is given by<br />
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H_0((V) = V^G.<br />
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(In other words: your job again is to explain how to associate to a morphism from (V,rho) to (W,phi) a morphism from V^G to W^G, in a way that's compatible with composition.)<br />
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<!--4. The group PSL_2(Z) has a presentation of the form [U,V|U^2 = V^3 = 1]. Thus, an n-dimensional representation of PSL_2(Z) is precisely a pair of n x n matrices (rho(U),rho(V)), such that rho(U)^2 = rho(V)^3 = 1. Prove that for every n > 1 there are uncountably many isomorphism classes of n-dimensional representations of PSL_2(Z). (Extra credit: show that there are uncountably many isomorphism classes of _irreducible_ representations.)<br />
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An amazing theorem of Margulis (superrigidity) shows that SL_2(Z) is very special among arithmetic groups in this way. For instance, there are only _finitely_ many isomorphism classes of n-dimensional representations of PSL_d(Z) for any d > 2; those groups are "rigid."--><br />
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<!--<br />
5. Let G be a finite group, let X be a set with G-action, and let chi be the character of the corresponding permutation representation. Show that, for each g in G, chi(g) is the number of elements of X fixed by g. (I am actually going to prove this in class.)<br />
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6. Let V be the space of homogeneous degree-3 polynomials in variables x_1, x_2, x_3; then S_3 acts on V by permutation of the three variables. Describe the decomposition of V into irreducible representations of S_3.<br />
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7a. Show that if chi is the character of a representation of a finite group G, then chi(g^{-1}) is the complex conjugate of chi.<br />
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7b. As a consequence, show that if g is conjugate to g^{-1} for all g in G, then chi(g) is real.<br />
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7c. The groups D_p and S_n have the property that g is conjugate to g^{-1} for _every_ g in G, so that every character is a real-valued function on G. Give an example of a group G where this is not the case, and give an example of a representation of G whose character takes non-real values.<br />
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==HOMEWORK 7 (due Nov 6)==<br />
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1. Let G be a finite group, let X be a finite set with G-action, and let chi be the character of the corresponding permutation representation. Show that, for each g in G, chi(g) is the number of elements of X fixed by g. (I am actually going to prove this in class.)<br />
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2a. Show that if chi is the character of a finite-dimensional representation of a finite group G, then chi(g^{-1}) is the complex conjugate of chi(g) for all g.<br />
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2b. As a consequence, show that if g is conjugate to g^{-1} for all g in G, then chi(g) is real.<br />
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2c. The groups D_p and S_n have the property that g is conjugate to g^{-1} for _every_ g in G, so that every character is a real-valued function on G. Give an example of a group G where this is not the case, and give an example of a representation of G whose character takes non-real values.<br />
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3. Let V and W be complex vector spaces. Let f be a linear transformation of V, and g a linear transformation of W. <br />
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3a. Show that there is a unique linear transformation F satisfying<br />
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F(v tensor w) = f(v) tensor g(w)<br />
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for all v in V and all w in W. We denote this transformation by f tensor g.<br />
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3b. Suppose V and W are finite dimensional. Show that Trace(f tensor g) = Trace(f) Trace(g). What does this say when f and g are both the identity transformation?<br />
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4. Let V be a vector space. We define Sym^2 V to be the quotient of V tensor V by the subspace generated by all elements of the form<br />
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v tensor w - w tensor v<br />
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for v,w in V.<br />
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Suppose dim V = n. What is dim Sym^2 V?<br />
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5. Show that Z/pZ tensor Z/qZ is zero when p and q are distinct primes.<br />
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==HOMEWORK 8 (due Nov 13)==<br />
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1. (Schur's lemma can be false over non-algebraically closed fields) Let G be Z/3Z and let g be a generator of G. Then G acts by cyclic permutations on the set {1,2,3}; let P be the corresponding permutation representation on the REAL (not complex!) vector space generated by e_1,e_2,e_3.<br />
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1a. Show that P breaks up as the direct sum of the trivial representation and a 2-dimensional representation V which is irreducible. (Remember, we are not working over the complex numbers, so you cannot use the criterion for irreducibility in terms of the character that we worked out in class for the complex case; you need to prove directly that V has no nontrivial proper subrepresentation.)<br />
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1b. Show that the map from V to V induced by g is G-equivariant, but is neither zero nor a scalar multiplication.<br />
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2. If V_1 and V_2 are representations (not necessarily irreducible!) of G, show that <chi_V_1, chi V_2> is a non-negative integer.<br />
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3. When a formula always yields a non-negative integer, it is often because it is secretly a formula for either the cardinality of some finite set or the dimension of some finite-dimensional vector space. In this case, show that<br />
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<chi_{V_1}, chi_{V_2}> = dim_C Hom_G(V_1,V_2)<br />
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where we recall that Hom_G(V_1,G_2) refers to the space of G-equivariant homomorphisms from V_1 to V_2.<br />
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4. If A is an abelian group, show that every irreducible representation of A is 1-dimensional. (Hint: show that there is a vector which is an eigenvector for every element of A!)<br />
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5. Let G be the group of order 20 which is the semidirect product of A by H, where A is Z/5Z and H is (Z/5Z)^* with its natural action on A. We will work out all the irreducible representations of A. <br />
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5a. First of all, consider the 5-dimensional permutation representation of G acting by left multiplication on the cosets of H. Show that this representation decomposes as a direct sum of the trivial representation and an irreducible 4-dimensional representation. (Hint: it is probably easier to use the character of this representation than to prove irreducibility directly.) <br />
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5b. Now show that there are 4 1-dimensional representations of G which factor through the quotient G/A (i.e. such that rho(A) is the identity.)<br />
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5c. Using the decomposition of the regular representation, show that the irreducible representations of G you have constructed are the only ones.<br />
<br />
==HOMEWORK 9 (due Nov 20)==<br />
<br />
0. Find all ring homomorphisms from Z to Z/30Z. [Warning: there are 8 of them, not 30.]<br />
<br />
<br />
1. If R is a ring (not necessarily with 1) such that a^2 = a for every a in R, R is called a Boolean ring.<br />
<br />
1a. Show that any nonzero Boolean ring has characteristic 2. (In other words, show that a+a=0 for all a in R.)<br />
<br />
1b. Show that every Boolean ring is commutative.<br />
<br />
<br />
2. Let phi : R -> S be a homomorphism of commutative rings with 1.<br />
<br />
2a. If I is an ideal of S, show that phi^(-1) (I) is an ideal of R.<br />
<br />
2b. If P is a prime ideal of S, show that phi^(-1) (P) is either R or a prime ideal of R.<br />
<br />
2c. If M is a maximal ideal of S and phi is surjective, show that phi^(-1) (M) is a maximal ideal of R.<br />
<br />
2d. Find a homomorphism phi : R -> S and a maximal ideal M of S such that phi^(-1) (M) is not a maximal ideal of R.<br />
<br />
<br />
3. Let Nil(A) denote the set of nilpotent elements of a ring A, and let R be a commutative ring. In class it was observed that Nil(R) is an ideal of R.<br />
<br />
3a. Find (a generator of) Nil(Z/720Z).<br />
<br />
3b. Show that Nil(R/Nil(R)) = 0.<br />
<br />
3c. If x is in Nil(R) and u is a unit, show that u+x is a unit.<br />
<br />
3d. Show that Nil(S), where S is the 2x2 matrices with real coefficients, is not an ideal of S.<br />
<br />
<br />
4a. If M is an R-module and I is an ideal of R such that im = 0 for all i in I and m in M, show that M carries the structure of an R/I-module.<br />
<br />
4b. If S is a commutative ring with 1 and J is a maximal ideal of S, show that J/J^2 is a vector space over S/J.<br />
<br />
<br />
5. A "divisible element" x of a Z-module is one such that for each positive integer k, there exists a y such that ky=x. Show that a free Z-module cannot contain a nonzero divisible element.<br />
<br />
<br />
6. Prove that the direct sum of any collection of free R-modules is also a free R-module.<br />
<br />
<br />
7. In this problem you will prove that the module M given by the direct product of copies of Z, indexed by the positive integers, is not a free Z-module. Therefore, assume that M is a free Z-module with basis B, and let N be the direct sum of copies of Z, as a submodule of M.<br />
<br />
7a. Show there is a countable subset B1 of B such that N is contained in the submodule N1 generated by B1. [Hint: N is countable.]<br />
<br />
7b. Show that M/N1 is a free Z-module.<br />
<br />
7c. Let S = {(b1, b2, b3, ...) : b_i = i! or -i!}. Show that there is some s in S which is not in N1. [Hint: S is uncountable.]<br />
<br />
7d. For any s in S but not N1, show that for each positive integer k, there exists some m in M with s-km in N1. Conclude that s+N1 is a divisible element of M/N1.<br />
<br />
7e. Use problem 5 to obtain a contradiction.<br />
<br />
<br />
8. In class I said "rank" is not always well-defined for R-modules if R is not commutative. Here is an example demonstrating this assertion. Let M be the countably infinite direct product of copies of Z as in exercise 7, and let R be the endomorphism ring of M. Define phi_1(a1,a2,a3,a4,...) = (a1,a3,a5,...) and phi_2(a1,a2,a3,a4,...) = (a2,a4,a6,...).<br />
<br />
8a. Show that {phi_1, phi_2} is a free basis for R, as a left R-module. [Hint: let psi_1(a1,a2,a3,...) = (a1,0,a2,0,...) and psi_2(a1,a2,a3,...) = (0,a1,0,a2,...). Compute the products of the phi_i and psi_j in either order, and use the result to show that the phi_i are a basis.]<br />
<br />
8b. Show that R is isomorphic to R^n for every positive integer n.<br />
<br />
<br />
==HOMEWORK 10 (due Dec 4)==<br />
<br />
1. '''How algebraists do calculus'''. The ''ring of dual numbers'' is the ring R = C[e]/(e^2). We think of e as an infinitesimal number; that is, something so small that its square is 0. This algebraic notion of infinitesimal is the basis for the algebraic notion of differentiation, which works as follows. Let f be a polynomial in R[x]. Show that there is a unique polynomial g in C[x] such that<br />
<br />
f(x+e) - f(x) = eg(x)<br />
<br />
and that in fact g is the derivative of f. In other words, the usual definition of "derivative" works just fine in this context, without any use of the notion of limit!<br />
<br />
2. Write down all the ideals of C[e]/e^2. (There are three.)<br />
<br />
3. Show that any right ideal of the ring M_2(Q) of 2x2 matrices is an ideal of the form I_V described in class for some subspace V of Q^2; that is, it consists precisely of those linear transformations whose image is contained in V.<br />
<br />
<br />
4. An ''idempotent'' element of a ring is an element e satisfying e^2 = e.<br />
<br />
4a. If e is an idempotent, show that 1-e is also an idempotent.<br />
<br />
4b. Suppose R has a unit (we are actually always supposing this, but I just want to remind you of it for this problem) and suppose e is a ''central'' idempotent, i.e. an idempotent which commutes with all elements of R. Show that eR and (1-e)R are both two-sided ideals. Moreover, show that eR and (1-e)R are also subrings of R (with the warning that the identity in eR is not 1, which is not contained in eR, but rather e.)<br />
<br />
4c. Show that the map f_e: R -> eR defined by f_e(r) = er is a ring homomorphism. Finally show that the ring homomorphism<br />
<br />
f_{e} x f_{1-e}: R -> eR x (1-e)R <br />
<br />
is actually an isomorphism of rings. In other words, the presence of a central idempotent induces a decomposition a ring as a direct product.<br />
<br />
<br />
5. The ''center'' of a ring R is the subring of elements commuting with every element of R.<br />
<br />
5a. Show that the center of M_n(Q) is the ring of scalar matrices.<br />
<br />
5b. Show that the center of the group ring C[G] is the set of elements sum f(g) g such that f(g) is a class function.<br />
<br />
<br />
6. '''How algebraists do differential equations.''' We define the ''Weyl algebra'' W of differential operators in one variable as follows. Multiplication by x is a linear transformation on the complex vector space V of all smooth functions on C. Differentiation, which we denote d, is also a linear transformation on V. So we let W be the subring of End(V) generated by x and d. We note that x and d satisfy the relation xd = dx - 1; in particular, the Weyl algebra is not commutative.<br />
<br />
6a. Describe the submodule of V which is annihilated by the element d-x in W.<br />
<br />
6b. Show that, for any polynomial f(x), fd-df = -f', and also show that for any polynomial g(d), gx-xg = -g'. (This is another way algebraists do calculus!) <br />
<br />
6c. Use the facts from (b) to show that the center of the Weyl algebra is just the field of constants. (Remark: you will use in the course of this proof the fact that a polynomial whose derivative is 0 is constant -- this would not be true if we were working in positive characteristic, as the example of x^p considered as a polynomial over F_p demonstrates.)<br />
<br />
<br />
7. If M is a left module for a ring R, we denote by Ann(M) the set of elements r in R such that rm = 0 for all m in M. Show that Ann(M) is a two-sided ideal. <br />
<br />
<br />
8. Let M be a left module for R, and let m be an element of M. We denote by Ann(m) the set of elements r in R such that rm = 0.<br />
<br />
8a. Show that Ann(m) is a left ideal, and show that the submodule Rm of M generated by m is isomorphic to the left module R/Ann(m).<br />
<br />
8c. Let V,W be as in question 6. Show that Ann(e^x) is the left ideal W(d-1).<br />
<br />
==HOMEWORK 11 (due Dec 13)==<br />
<br />
1. Let f: R -> S be a homomorphism of rings. As discussed in class, if M is a left R-module, then S tensor_R M is a left S-module, and if N is an S-module, then N can also be considered as an R-module by pullback of the action.<br />
<br />
Prove that if M is a finitely generated left R-module, then S tensor_R M is a finitely generated left S-module. On the other hand, give an example to show that N can be finitely generated as a left S-module but not finitely generated when considered as a left R-module.<br />
<br />
2. Recall from class that when V and W are complex vector spaces, W^dual tensor V is naturally isomorphic to the space of linear homomorphisms Hom(W,V).<br />
<br />
Now suppose that V and W are left C[G]-modules, i.e. complex representations of a group G. Show that W^dual tensor_C[G} V is the maximal G-invariant quotient of Hom(W,V); in other words, the quotient of Hom(W,V) by the subspace generated by all functions F of the form F(w) = f(gw) - gf(w) for some function f in Hom(W,V) and some g in G.<br />
<br />
3. '''How algebraists do logic.''' Why are Boolean rings called Boolean rings? Because they model some features of the Boolean arithmetic that governs propositional logic. Let A be a collection of propositions. We define two operations on A as follows. By a + b we mean the negation of a xor b, while by ab we mean "a or b."<br />
<br />
3a. It is not a priori obvious that these operations satisfy the ring axioms, but assume this for the moment. Show that a^2 = a for all a (so A is a boolean ring) and 2a = 0 for all a (so A has characteristic 2.)<br />
<br />
3b. Suppose we have a consistent assignment of truth-values to the propositions in A. Show that the set of true statements of A forms an ideal of A.<br />
<br />
3c. (Optional for people who like propositional logic) Show that a(b+c) = ab + ac with the given operations.<br />
<br />
<br />
4. Let K be Q[x]/(x^3-2).<br />
<br />
4a. Show that K is a field.<br />
<br />
4b. Show that K tensor_Q K decomposes as the direct sum of two fields. What are they? [Note: K tensor_Q K is a ring!]<br />
<br />
5. There was some discussion in class about when we can "go back and forth" between right modules and left modules, as is the case with group rings.<br />
<br />
5a. If R is a ring, we denote R^op ("the opposite ring") to be the ring whose elements are those of R, but whose multiplication law x_opp is defined by s x_opp r = rs. Show that if M is a left R-module, then the map M x R^op -> M sending (x,r) to rx makes M into a right R^op module. In particular, if there is a RING ISOMORPHISM from R to R^op, then every left R-module can be thought of as a right R-module.<br />
<br />
5b. Verify that the map sending g to g^{-1} extends by linearity to give a ring isomorphism between C[G] and C[G]^op.<br />
<br />
5c. Construct a ring isomorphism between M_n(Q) and M_n(Q)^op.<br />
<br />
5d. Optional: Give an example of a ring (with 1) which is not isomorphic to its opposite ring. (This is not so easy, I think! I found a MathOverflow thread with several examples, none of which came with a short proof of the non-isomorphism -- try your hand!)<br />
<br />
6. . Let R be a commutative ring with 1. If R has a unique maximal ideal M, R is called a "local ring".<br />
<br />
6a. If R is a local ring, show that every element of R that is not in M is a unit.<br />
<br />
6b. Conversely, if R is such that the set of nonunits forms an ideal I, show that R is a local ring with maximal ideal I.<br />
<br />
7. The inclusion of S_3 into S_4 associated to the inclusion of {1,2,3} into {1,2,3,4} induces a ring homomorphism from C[S_3} to C{S_4]. Let V be a representation of C{S_3]. Then, as discussed in class, V tensor_C[S_3] C[S_4] is a representation of S_4, called the '''induced representation''' of V.<br />
<br />
7a. Show that dim_C (V tensor_C[S_3] C[S_4]) = 4 dim_C V.<br />
<br />
7b. When V is the trivial representation of S_3, decompose (V tensor_C[S_3] C[S_4]) as a sum of irreducible representations of S_4.<br />
<br />
--><br />
<br />
<!--==HOMEWORK PROBLEMS TO GIVE IN FUTURE SEMESTERS==<br />
<br />
Let f: C[x] tensor_C C[y] -> C[z] be the map of rings with f(x) = f(y) = z. Show that f(m tensor n) is not 0 for any pure tensor m tensor n, but f is not injective. (This is a warning not to try to check injectivity by checking that no pure tensors are killed by the map.)<br />
<br />
--></div>Dummithttps://hilbert.math.wisc.edu/wiki/index.php?title=741&diff=60117412013-09-29T04:10:32Z<p>Dummit: </p>
<hr />
<div>'''Math 741'''<br />
<br />
Algebra<br />
<br />
Prof: [http://www.math.wisc.edu/~ellenber Jordan Ellenberg]<br />
<br />
Grader: Evan Dummit.<br />
* Homework policies:<br />
* Late homework may be given directly to the grader, along with either <br />
(i) the instructor's permission, or (ii) a polite request for mercy.<br />
* Assignments that are more than 1 page should be affixed in some reasonable way.<br />
* Results from places (e.g., the internet) other than 741 and standard books must be cited.<br />
<br />
JE's office hours: Monday 12pm-1pm (right after class)<br />
<br />
This course, the first semester of the introductory graduate sequence in algebra, will cover the basic theory of groups, group actions, representation, linear and multilinear algebra, and the beginnings of ring theory.<br />
<br />
==SYLLABUS==<br />
<br />
In this space we will record the theorems and definitions we covered each week, which we can use as a list of notions you should be prepared to answer questions about on the Algebra qualifying exam. The material covered on the homework is also an excellent guide to the scope of the course. <br />
<br />
'''WEEK 1''': <br />
<br />
Definition of group. Associativity. Inverse. <br />
<br />
Examples of group: GL_n(R). GL_n(Z). Z/nZ. R. Z. R^*. The free group F_k on k generators. <br />
<br />
Homomorphisms. The homomorphisms from F_k to G are in bijection with G^k. Isomorphisms.<br />
<br />
'''WEEK 2''':<br />
<br />
The symmetric group (or permutation group) S_n on n letters. Cycle decomposition of a permutation. Order of a permutation. Thm: every element of a finite group has finite order. <br />
<br />
Subgroups. Left and right cosets. Lagrange's Theorem. Cyclic groups. The order of an element of a finite group is a divisor of the order of the group.<br />
<br />
The sign homomorphism S_n -> +-1.<br />
<br />
<br />
'''WEEK 3'''<br />
<br />
Normal subgroups. The quotient of a group by a normal subgroup. The first isomorphism theorem. Examples of S_n -> +-1 and S_4 -> S_3 with kernel V_4, the Klein 4-group.<br />
<br />
Centralizers and centers. Abelian groups. The center of SL_n(R) is either 1 or +-1.<br />
<br />
Groups with presentations. The infinite dihedral group <x,y | x^2 = 1, y^2 = 1>.<br />
<br />
'''WEEK 4'''<br />
<br />
More on groups with presentations.<br />
<br />
Second and third isomorphism theorems.<br />
<br />
Semidirect products.<br />
<br />
'''WEEK 5'''<br />
<br />
Group actions, orbits, and stabilizers.<br />
<br />
Orbit-stabilizer theorem.<br />
<br />
Cayley's theorem.<br />
<br />
Cauchy's theorem.<br />
<br />
'''WEEK 6'''<br />
<br />
Applications of orbit-stabilizer theorem (p-groups have nontrivial center, first Sylow theorem.)<br />
<br />
Classification of finite abelian groups and finitely generated abelian groups.<br />
<br />
Composition series and the Jordan-Holder theorem (which we state but don't prove.) <br />
<br />
The difference between knowing the composition factors and knowing the group (e.g. all p-groups of the same order have the same composition factors.)<br />
<br />
'''WEEK 7'''<br />
<br />
Simplicity of A_n.<br />
<br />
Nilpotent groups (main example: the Heisenberg group)<br />
<br />
Derived series and lower central series.<br />
<br />
Category theory 101: Definition of category and functor. Some examples. A group is a groupoid with one object.<br />
<br />
'''WEEK 8'''<br />
<br />
Introduction to representation theory.<br />
<br />
'''WEEK 10'''<br />
<br />
Ring theory 101: Rings, ring homomorphisms, ideals, isomorphism theorems. Examples: fields, Z, the Hamilton quaternions, matrix rings, rings of polynomials and formal power series, quadratic integer rings, group rings. Integral domains. Maximal and prime ideals. The nilradical.<br />
<br />
Module theory 101: Modules, module homomorphisms, submodules, isomorphism theorems. Noetherian modules and Zorn's lemma. Direct sums and direct products of arbitrary collections of modules.<br />
<br />
<br />
==HOMEWORK 1 (due Sep 16)==<br />
<br />
1. Suppose that H_1 and H_2 are subgroups of a group G. Prove that the intersection of H_1 and H_2 is a subgroup of G.<br />
<br />
2. Recall that S_3 (the symmetric group) is the group of permutations of the set {1..3}. List all the subgroups of S_3.<br />
<br />
3. We can define an equivalence relation on rational numbers by declaring two rational numbers to be equal whenever they differ by an integer. We denote the set of equivalence classes by Q/Z. The operation of addition makes Q/Z into a group.<br />
<br />
a) For each n, prove that Q/Z has a subgroup of order n.<br />
<br />
b) Prove that Q/Z is a ''divisible'' group: that is, if x is an element of Q/Z and n is an integer, there exists an element y of Q/Z such that ny = x. (Note that we write the operation in this group as addition rather than multiplication, which is why we write ny for the n-fold product of y with itself rather than y^n)<br />
<br />
c) Prove that Q/Z is not finitely generated. (Hint: prove that if x_1, .. x_d is a finite subset of Q/Z, the subgroup of Q/Z generated by x_1, ... x_d is finite.)<br />
<br />
d) Conclude that Q is not finitely generated.<br />
<br />
4. We will prove that there is no homomorphism from SL_2(Z) to Z except the one which sends all of SL_2(Z) to 0. Suppose f is a homomorphism from SL_2(Z) to Z.<br />
<br />
a) Let U1 be the upper triangular matrix with 1's on the diagonal and a 1 in the upper right hand corner, as in class, and let U2 be the transpose of U1, also as in class. Show that (U1 U2^{-1})^6 = identity (JING 1, TAO 0) and explain why this implies that f(U1) = f(U2).<br />
<br />
b) Show that there is a matrix A in SL_2(Z) such that A U1 A^{-1} = U2^{-1}. (Recall that we say U1 and U2^{-1} are "conjugate".) Explain why this also implies that f(U1) = -f(U2).<br />
<br />
c) Explain why a) and b) imply that f must be identically 0.<br />
<br />
5. The argument above also shows that there is no nonzero homomorphism from SL_2(Z) to Z/pZ where p is a prime greater than 3. However, it leaves open the possibility that there is indeed a nonzero homomorphism from SL_2(Z) to Z/2Z. Exhibit such a homomorphism. Optional challenge problem: exhibit a nonzero homomorphism from SL_2(Z) to Z/3Z.<br />
<br />
<br />
==HOMEWORK 2 (due Sep 25)==<br />
<br />
1. Let V be the Klein 4-group in S_4. Let Q be the symmetric group on the set DF of double-flips in S_4; there are 3 double flips, so Q is isomorphic to S_3. If g is an element of S_4, we discussed in class that conjugation by g permutes the elements of DF. So to each element of g, we have associated an element f(g) of Q. More or less by definition, this defines a homomorphism f: G -> Q. Show that this homomorphism is surjective and has kernel H, and thus that G/H is isomorphic to Q.<br />
<br />
2. Let F_2 be the free group on two generators, which we denote x,y. Prove that there is a automorphism of F_2 which sends x to xyx and y to xy, and prove that this automorphism is unique.<br />
<br />
3. Let H be a subgroup of G, and let N_G(H), the ''normalizer'' of H in G, be the set of elements g in G satisfying g H g^{-1} = H. Prove that N_G(H) is a subgroup of G.<br />
<br />
4. Let T be the subgroup of GL_2(R) consisting of diagonal matrices. This is an example of a "Cartan subgroup," or "torus" (whence the notation T.) Describe the normalizer N(T) of T in GL_2(R). Prove that there is a homomorphism from N(T) to Z/2Z whose kernel is precisely T. <br />
<br />
5. Prove that the group of inner automorphisms of a group is normal in the full automorphism group Aut(G).<br />
<br />
6. Let H and H' be subgroups of a group G, and let x be an element of G. We denote by HxH' the set of elements of the form hxh', with h in H and h' in H', and we call HxH a ''double coset'' of the pair (H,H').<br />
<br />
a) Show that G decomposes as a disjoint union of double cosets of (H,H').<br />
<br />
b) Let G be GL_2(R) and let B be the subgroup of upper triangular matrices. Prove that G decomposes into exactly two double cosets of (B,B). (This is an example of the so-called *Bruhat decomposition* which is of great importance in the theory of algebraic groups and their representations.)<br />
<br />
c) Let G be the symmetric group S_n, and let H be the subgroup of permutations which fix 1. Describe the double coset decomposition of S_n into double cosets of (H,H).<br />
<br />
<br />
==HOMEWORK 3 (due Oct 2)==<br />
<br />
1. Compute the centralizer of the triple flip (12)(34)(56) in S_6. (Note: this is the kind of problem that a computer algebra package like sage is very good at, and I encourage you to learn to use sage well enough to check your answer.)<br />
<br />
2. Warning: a normal subgroup of a normal subgroup need not be normal! Let H be the subgroup of S_4 generated by (12)(34). Then H is a subgroup of the Klein 4-group V_4, which we have already shown is normal in S_4. Show that H is normal in V_4, but H is not normal in S_4.<br />
<br />
3. Let G be a group and let H_1 and H_2 be subgroups of G of finite index (that is, the number of cosets in G/H_i is finite, i = 1,2.) Prove that H_1 intersect H_2 also has finite index in G.<br />
<br />
4. Let X be the set of lines in F_3^2 (here a line means a 1-dimensional linear subspace) so that |X| = 4. Let G be the projective linear group PGL_2(F_3), so that G acts on X via its action on F_3^2. This action can be thought of as a homomorphism from G to S_4. Show that this homomorphism is an ISOMORPHISM from PGL_2(F_3) to S_4.<br />
<br />
5. We proved two theorems in class: Lagrange's theorem, which says that the order of a subgroup of G divides |G|, and Cauchy's theorem, which says that if p divides |G| then there exists a subgroup of G of order p. Cauchy's theorem is a kind of partial converse to Lagrange's theorem, but the full converse is false. Give an example of a finite group G and a divisor n of |G| such that you can prove there is no subgroup of G of order n.<br />
<br />
6. The group S_3 has only three conjugacy classes. Prove that a finite group with at most three conjugacy classes has order at most 6.<br />
<br />
7. Let X be the set of ordered triples of elements of {1,..,n}, for some n >= 3. Then S_n acts on X. How many orbits are there? (Hint: the answer does not depend on n.) OPTIONAL: How many orbits are there on the space of ordered k-tuples, when n >= k? <br />
<br />
<br />
<!--Below you will find a repository of homework problems. Note that some of these problems are taken from Lang's ''Algebra''.<br />
<br />
==HOMEWORK 1 (due Sep 20)==<br />
<br />
1. Suppose that H_1 and H_2 are subgroups of a group G. Prove that the intersection of H_1 and H_2 is a subgroup of G.<br />
<br />
2. Recall that S_3 (the symmetric group) is the group of permutations of the set {1..3}. List all the subgroups of S_3.<br />
<br />
3. We can define an equivalence relation on rational numbers by declaring two rational numbers to be equal whenever they differ by an integer. We denote the set of equivalence classes by Q/Z. The operation of addition makes Q/Z into a group.<br />
<br />
a) For each n, prove that Q/Z has a subgroup of order n.<br />
<br />
b) Prove that Q/Z is a ''divisible'' group: that is, if x is an element of Q/Z and n is an integer, there exists an element y of Q/Z such that ny = x. (Note that we write the operation in this group as addition rather than multiplication, which is why we write ny for the n-fold product of y with itself rather than y^n)<br />
<br />
c) Prove that Q/Z is not finitely generated. (Hint: prove that if x_1, .. x_d is a finite subset of Q/Z, the subgroup of Q/Z generated by x_1, ... x_d is finite.)<br />
<br />
4. Let F_2 be the free group on two generators x,y. Prove that there is a automorphism of F_2 which sends x to xyx and y to xy, and prove that this automorphism is unique.<br />
<br />
5. Let H be a subgroup of G, and let N_G(H), the ''normalizer'' of H in G, be the set of elements g in G satsifying g H g^{-1} = H. Prove that N_G(H) is a subgroup of G.<br />
<br />
6. Let T be the subgroup of GL_2(R) consisting of diagonal matrices. This is an example of a "Cartan subgroup," or "torus" (whence the notation T.) Describe the normalizer N(T) of T in GL_2(R). Prove that there is a homomorphism from N(T) to Z/2Z whose kernel is precisely T. <br />
<br />
7. Prove that the group of inner automorphisms of a group is normal in the full automorphism group Aut(G).<br />
<br />
8. Let H and H' be subgroups of a group G, and let x be an element of G. We denote by HxH' the set of elements of the form hxh', with h in H and h' in H', and we call HxH a ''double coset'' of the pair (H,H').<br />
<br />
a) Show that G decomposes as a disjoint union of double cosets of (H,H').<br />
<br />
b) Let G be GL_2(R) and let B be the subgroup of upper triangular matrices. Prove that G decomposes into exactly two double cosets of (B,B).<br />
<br />
c) Let G be the symmetric group S_n, and let H be the subgroup of permutations which fix 1. Describe the double coset decomposition of S_n into double cosets of (H,H).<br />
<br />
==HOMEWORK 2 (due Sep 27)==<br />
<br />
1. Prove that the symmetric group S_n is generated by transpositions. Hint: proceed by induction on n. Let H be the subgroup of S_n consisting of permutations fixing 1. Observe that H is isomorphic to S_{n-1}, so that H is generated by transpositions. Explain why it suffices to show that, for any g in S_n, the coset gH contains a transposition. Then explain why this is the case.<br />
<br />
2. Given that the symmetric group S_n is generated by transpositions, show that there are no nontrivial homomorphisms from S_n to Z/pZ for any odd prime p.<br />
<br />
3. Compute the centralizer of the triple flip (12)(34)(56) in S_6. (Note: this is the kind of problem that a computer algebra package like sage is very good at, and I encourage you to learn to use sage well enough to check your answer.)<br />
<br />
4. Warning: a normal subgroup of a normal subgroup need not be normal! Let H be the subgroup of S_4 generated by (12)(34). Then H is a subgroup of the Klein 4-group V_4, which we have already shown is normal in S_4. Show that H is normal in V_4, but H is not normal in S_4.<br />
<br />
5. The '''quaternion group''' Q is a non-abelian finite group of order 8. Its elements are 1, -1, i, -i, j, -j, k, -k; it satisfies g^2 = -1 for all g except +-1, and ijk = 1. (Note that the notation "-i" means "the product of i with -1" and -1 is understood to be central in Q.) Prove that every subgroup of Q is normal.<br />
<br />
6. (finishing example done in class) Let Gamma be the group F<x,y> / (x^2 = 1, y^2 = 1). (Remember, this means the quotient of the free group by the group generated by all conjugates of x^2 and y^2.) Prove that Gamma is infinite. Prove that Gamma has a subgroup R generated by xy isomorphic to Z, that R is normal, and that the quotient Gamma/R is isomorphic to Z/2Z. <br />
<br />
7. Suppose a group G has the property that every element in g satisfies g^2 = 1. Show that G must be abelian.<br />
<br />
==HOMEWORK 3 (due Oct 4)==<br />
<br />
1. Multiplication by -1 is an automorphism of Z/nZ. Let a be the homomorphism Z/2Z -> Aut(Z/nZ) which sends the nontrivial element of Z/2Z to multiplication by -1. Then we have a semidirect product of Z/pZ by Z/2Z as defined in class. This group of order 2n is called a ''dihedral group.'' and is denoted D_n (or sometimes D_{2n}).<br />
<br />
1a. Compute the center of D_n. (Note that the answer depends on n!)<br />
1b. Show that all involutions of D_n are conjugate to each other if and only if n is odd.<br />
<br />
2. (More on direct products that we didn't do in class.) Suppose that G is a semidirect product of N with H. Suppose furthermore that H is also normal in G. Prove that G is a direct product of N with H.<br />
<br />
3. Let Q be the quaternion group from last week's homework. Show that Q does not decompose as a semidirect product N \rtimes H, apart form the trivial cases N = {e} and N = Q.<br />
<br />
4. The ''affine linear group'' of degree n is the group of transformations from R^n to R^n of the form x -> Ax + b, where A is a matrix in GL_n(R) and b is a vector in R^n. Prove that the affine linear group is a semidirect product R^n \rtimes GL_n.<br />
<br />
5. The ''ordinary triangle group'' T(p,q,r) is the group with presentation <x,y | x^p = y^q = (xy)^r = 1>. This is a very interesting family of groups which arises naturally in many contexts in number theory and topology.<br />
<br />
6a. Prove that T(2,2,n) is isomorphic to the dihedral group D_n.<br />
6b. Prove that T(2,3,3) is finite and compute its order. Is it isomorphic to a finite group we've discussed in class?<br />
<br />
(It is a beautiful fact, but outside the scope of this course, that T(p,q,r) is finite if and only if 1/p + 1/q + 1/r > 1.)<br />
<br />
7. Let G be a subgroup of GL_n(R) which contains SL_n(R). Prove that G is a normal subgroup of GL_n(R).<br />
<br />
8. Let G be a group and let H_1 and H_2 be subgroups of G of finite index (that is, the number of cosets in G/H_i is finite, i = 1,2.) Prove that H_1 intersect H_2 also has finite index in G.<br />
<br />
9. Recall that the commutator [x,y] is x y x^{-1} y^{-1}. If G is a group, the commutator subgroup of G, denoted G' or [G,G], is the subgroup of G generated by all commutators [x,y] for x,y in G.<br />
<br />
9a. Show that G' is a normal subgroup of G.<br />
9b. Show that G/G' is an abelian group.<br />
9c. Show that if f: G -> A is a homomorphism, with A abelian, then G' is contained in ker f. Conclude that f factors through a homomorphism G/G' -> A.<br />
9d. Show that G' = G if and only if G has no nontrivial abelian quotient. In this case, we say G is ''perfect''.<br />
<br />
10. Under what conditions on (p,q,r) is the triangle group T(p,q,r) perfect?<br />
<br />
==HOMEWORK 4 (due Oct 16)==<br />
<br />
1. Let X be a set with a transitive action of a group G, which we think of as a homomorphism f: G -> Sym(X). Let H_x be the stabilizer of an element x of X.<br />
<br />
1a. If x' is another element of X, show that H_{x"} and H_x are conjugate subgroups. (Hint: this is not true without the hypothesis of transitivity, so make sure to use it!)<br />
<br />
1b. Show that the kernel of f is the intersection of all the conjugates of H_x in G. (This subgroup is called the ''normal core'' of H_x.)<br />
<br />
2. This week, a certified mathematical grown-up asked on MathOverflow what the index-3 subgroups of SL_2(Z/pZ) were. Let's do it ourselves for homework. Let p be a prime, and suppose H is an index-3 subgroup of SL_2(Z/pZ). I do not claim that the proof sketched below is necessarily the easiest, but it allows us to cover some ground that we won't get to do in lecture.<br />
<br />
2a. Using the coset action, show that the normal core of H is the kernel of a homomorphism f from SL_2(Z/pZ) to S_3.<br />
<br />
2b. A ''unipotent matrix'' is a matrix A such that (A-I)^n = 0 for some positive integer n. Prove that a unipotent element of SL_2(Z/pZ) has order dividing p.<br />
<br />
2c. You may use the fact that unipotent matrices generate SL_2(Z/pZ). Given this fact, show that f must be trivial when p > 3.<br />
<br />
2d. Using the above, show that SL_2(Z/pZ) has no index-3 subgroups when p > 3.<br />
<br />
2e. (extra credit) What can you say about the minimal index of a proper subgroup of SL_2(Z/pZ)?<br />
<br />
3. Let H be a subgroup of G of index 2. Prove that H is normal.<br />
<br />
4. We proved two theorems in class: Lagrange's theorem, which says that the order of a subgroup of G divides |G|, and Cauchy's theorem, which says that if p divides |G| then there exists a subgroup of G of order p. Cauchy's theorem is a kind of partial converse to Lagrange's theorem, but the full converse is false. Give an example of a finite group G and a divisor n of |G| such that you can prove there is no subgroup of G of order n.<br />
<br />
5. Let X be the set of lines in F_3^2 (here a line means a 1-dimensional linear subspace) so that |X| = 4. Let G be the projective linear group PGL_2(F_3), so that G acts on X via its action on F_3^2. This action can be thought of as a homomorphism from G to S_4. Show that this homomorphism is an ISOMORPHISM from PGL_2(F_3) to S_4.<br />
<br />
6. The group S_3 has only three conjugacy classes. Prove that a finite group with at most three conjugacy classes has order at most 6.<br />
<br />
7. Let X be the set of ordered triples of elements of {1,..,n}, for some n >= 3. Then S_n acts on X. How many orbits are there? (Hint: the answer does not depend on n.)<br />
<br />
8. Suppose that G and H are groups and f: G -> H is a homomorphism. Recall the definition of the abelianization G^ab from the previous problem set.<br />
<br />
8a. Show that the composition G -> H -> H^ab factors through a unique homomorphism G^ab -> H^ab, which we denote f^ab.<br />
<br />
8b. Show that if f: G -> H and g: H -> Q are homomorphisms, then f^ab o g^ab = (f o g)^ab.<br />
<br />
(For those reading MacLane, this constitutes a proof that abelianization is a '''functor''' from the category of groups to the category of abelian groups.)<br />
<br />
==HOMEWORK 5 (due Oct 23)==<br />
<br />
1. Show that there are only two different isomorphism classes of groups of order 14. (Hint: prove that such a group must be a semidirect product of a group of order 7 by a group of order 2.)<br />
<br />
2. (variant of Burnside's Lemma) Let G be a finite group and H a normal subgroup. Suppose G acts on a finite set X. <br />
<br />
2a. Show that if x and y lie in the same H-orbit, and g is an element of G, then gx and gy also lie in the same H-orbit. Thus, G/H has a natural action on the set of H-orbits X/H.<br />
<br />
2b. Show that the average number of fixed points in X of an element of the coset gH is the same as the number of fixed points of the element gH of G/H in its action on X/H.<br />
<br />
3. A ''central extension'' of a group G by an abelian group A is a group E, together with a surjective homomorphism E -> G whose kernel is central in E and is isomorphic to A.<br />
<br />
3a. Show that the quaternion group is a central extension of (Z/2Z)^2 by Z/2Z.<br />
<br />
3b. Give an example of another nonabelian group which is a central extension of (Z/2Z)^2 by Z/2Z and which is not isomorphic to the quaternion group.<br />
<br />
4. Prove that a central extension of an abelian group is nilpotent.<br />
<br />
5. Give two different composition series for S_4 and show that they have the same composition factors.<br />
<br />
6. Suppose that A is a finitely generated abelian group, and B is a subgroup of A.<br />
<br />
6a. Prove that B is finitely generated and that rank(B) <= rank(A).<br />
<br />
6b. Prove that rank(A/B) = rank(A) - rank(B).<br />
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7. For this problem, you may use the fact (which I should have asked on homework earlier) that if H_1 and H_2 are subgroups of G, then<br />
<br />
[G: H_1 intersect H_2] <= [G:H_1][G:H_2].<br />
<br />
Prove that if G is a simple group, d is an integer greater than 1, and S is a subgroup of index d in G, then |G| <= d^d.<br />
<br />
Conclude that, in particular, A_5 has no subgroup of index 3. On the other hand, exhibit a subgroup of A_5 of index 5.<br />
<br />
(Note: d^d is not a sharp bound; can you get a bound of d factorial instead?)<br />
<br />
<!--(This argument shows that a solvable subgroup of A_n has size at most about (n-1)!, but this is far from sharp; e.g. I found a citation to a 1967 paper of Dixon which shows that a solvable subgroup of A_n has size at most k^n where k is around 2.88.)--><br />
<br />
<!--<br />
8. Give an example of an abelian group A with a subgroup B such that B is not a direct summand of A.<br />
<br />
9. Let A be the subgroup of Z^2 generated by the vector (1,0). Let B be the subgroup of Z^2 generated by the vector (3,3). Show that A and B are isomorphic to each other, but Z^2 / A is not isomorphic to Z^2 / B.<br />
<br />
==HOMEWORK 6 (due Oct 30)==<br />
<br />
1. In this problem, you will construct a functor F from the category of finite sets to the category of complex vector spaces. I'll tell you what it does on objects: if X is a finite set, then F(X) is the vector space spanned by a set of basis elements {e_x}_{x in X}, so that dim F(X) = |X|. Your job: complete the definition of F by describing the map from Mor_{Finite Sets}(X,Y) to Mor_{Vector Spaces}(F(X),F(Y)), and showing that your map respects composition.<br />
<br />
2. If (V, rho) is a representation of a group G, denote by V^G the space of vectors in V such that gv = v for all g in G; these are called _invariant_ vectors. We saw that the permutation representation of S_3 has a one-dimensional space of invariants. Prove that if X is a finite set with G-action, and V_X the corresponding permutation representation of G, then<br />
<br />
dim V_X^G = number of orbits of X.<br />
<br />
3. (Invariants are functorial) Let G be a group. Then there is a category called C[G]-Mod whose objects are complex representations of G (i.e. pairs (V,rho) where V is a complex vector space and rho is a homomorphism from G to GL(V)). The morphisms in this category are the ones described in class: Mor((V,rho),(W,psi)) is the set of linear maps f from V to W such that<br />
<br />
f(rho(g)(v)) = psi(g)(f(v))<br />
<br />
for all g in G and all v in V.<br />
<br />
Show that there is a functor H_0 from the category of representations of G to the category of vector spaces whose action on objects is given by<br />
<br />
H_0((V) = V^G.<br />
<br />
(In other words: your job again is to explain how to associate to a morphism from (V,rho) to (W,phi) a morphism from V^G to W^G, in a way that's compatible with composition.)<br />
<br />
<!--4. The group PSL_2(Z) has a presentation of the form [U,V|U^2 = V^3 = 1]. Thus, an n-dimensional representation of PSL_2(Z) is precisely a pair of n x n matrices (rho(U),rho(V)), such that rho(U)^2 = rho(V)^3 = 1. Prove that for every n > 1 there are uncountably many isomorphism classes of n-dimensional representations of PSL_2(Z). (Extra credit: show that there are uncountably many isomorphism classes of _irreducible_ representations.)<br />
<br />
An amazing theorem of Margulis (superrigidity) shows that SL_2(Z) is very special among arithmetic groups in this way. For instance, there are only _finitely_ many isomorphism classes of n-dimensional representations of PSL_d(Z) for any d > 2; those groups are "rigid."--><br />
<br />
<!--<br />
5. Let G be a finite group, let X be a set with G-action, and let chi be the character of the corresponding permutation representation. Show that, for each g in G, chi(g) is the number of elements of X fixed by g. (I am actually going to prove this in class.)<br />
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6. Let V be the space of homogeneous degree-3 polynomials in variables x_1, x_2, x_3; then S_3 acts on V by permutation of the three variables. Describe the decomposition of V into irreducible representations of S_3.<br />
<br />
7a. Show that if chi is the character of a representation of a finite group G, then chi(g^{-1}) is the complex conjugate of chi.<br />
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7b. As a consequence, show that if g is conjugate to g^{-1} for all g in G, then chi(g) is real.<br />
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7c. The groups D_p and S_n have the property that g is conjugate to g^{-1} for _every_ g in G, so that every character is a real-valued function on G. Give an example of a group G where this is not the case, and give an example of a representation of G whose character takes non-real values.<br />
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==HOMEWORK 7 (due Nov 6)==<br />
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1. Let G be a finite group, let X be a finite set with G-action, and let chi be the character of the corresponding permutation representation. Show that, for each g in G, chi(g) is the number of elements of X fixed by g. (I am actually going to prove this in class.)<br />
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2a. Show that if chi is the character of a finite-dimensional representation of a finite group G, then chi(g^{-1}) is the complex conjugate of chi(g) for all g.<br />
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2b. As a consequence, show that if g is conjugate to g^{-1} for all g in G, then chi(g) is real.<br />
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2c. The groups D_p and S_n have the property that g is conjugate to g^{-1} for _every_ g in G, so that every character is a real-valued function on G. Give an example of a group G where this is not the case, and give an example of a representation of G whose character takes non-real values.<br />
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3. Let V and W be complex vector spaces. Let f be a linear transformation of V, and g a linear transformation of W. <br />
<br />
3a. Show that there is a unique linear transformation F satisfying<br />
<br />
F(v tensor w) = f(v) tensor g(w)<br />
<br />
for all v in V and all w in W. We denote this transformation by f tensor g.<br />
<br />
3b. Suppose V and W are finite dimensional. Show that Trace(f tensor g) = Trace(f) Trace(g). What does this say when f and g are both the identity transformation?<br />
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4. Let V be a vector space. We define Sym^2 V to be the quotient of V tensor V by the subspace generated by all elements of the form<br />
<br />
v tensor w - w tensor v<br />
<br />
for v,w in V.<br />
<br />
Suppose dim V = n. What is dim Sym^2 V?<br />
<br />
5. Show that Z/pZ tensor Z/qZ is zero when p and q are distinct primes.<br />
<br />
==HOMEWORK 8 (due Nov 13)==<br />
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1. (Schur's lemma can be false over non-algebraically closed fields) Let G be Z/3Z and let g be a generator of G. Then G acts by cyclic permutations on the set {1,2,3}; let P be the corresponding permutation representation on the REAL (not complex!) vector space generated by e_1,e_2,e_3.<br />
<br />
1a. Show that P breaks up as the direct sum of the trivial representation and a 2-dimensional representation V which is irreducible. (Remember, we are not working over the complex numbers, so you cannot use the criterion for irreducibility in terms of the character that we worked out in class for the complex case; you need to prove directly that V has no nontrivial proper subrepresentation.)<br />
<br />
1b. Show that the map from V to V induced by g is G-equivariant, but is neither zero nor a scalar multiplication.<br />
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2. If V_1 and V_2 are representations (not necessarily irreducible!) of G, show that <chi_V_1, chi V_2> is a non-negative integer.<br />
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3. When a formula always yields a non-negative integer, it is often because it is secretly a formula for either the cardinality of some finite set or the dimension of some finite-dimensional vector space. In this case, show that<br />
<br />
<chi_{V_1}, chi_{V_2}> = dim_C Hom_G(V_1,V_2)<br />
<br />
where we recall that Hom_G(V_1,G_2) refers to the space of G-equivariant homomorphisms from V_1 to V_2.<br />
<br />
4. If A is an abelian group, show that every irreducible representation of A is 1-dimensional. (Hint: show that there is a vector which is an eigenvector for every element of A!)<br />
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5. Let G be the group of order 20 which is the semidirect product of A by H, where A is Z/5Z and H is (Z/5Z)^* with its natural action on A. We will work out all the irreducible representations of A. <br />
<br />
5a. First of all, consider the 5-dimensional permutation representation of G acting by left multiplication on the cosets of H. Show that this representation decomposes as a direct sum of the trivial representation and an irreducible 4-dimensional representation. (Hint: it is probably easier to use the character of this representation than to prove irreducibility directly.) <br />
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5b. Now show that there are 4 1-dimensional representations of G which factor through the quotient G/A (i.e. such that rho(A) is the identity.)<br />
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5c. Using the decomposition of the regular representation, show that the irreducible representations of G you have constructed are the only ones.<br />
<br />
==HOMEWORK 9 (due Nov 20)==<br />
<br />
0. Find all ring homomorphisms from Z to Z/30Z. [Warning: there are 8 of them, not 30.]<br />
<br />
<br />
1. If R is a ring (not necessarily with 1) such that a^2 = a for every a in R, R is called a Boolean ring.<br />
<br />
1a. Show that any nonzero Boolean ring has characteristic 2. (In other words, show that a+a=0 for all a in R.)<br />
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1b. Show that every Boolean ring is commutative.<br />
<br />
<br />
2. Let phi : R -> S be a homomorphism of commutative rings with 1.<br />
<br />
2a. If I is an ideal of S, show that phi^(-1) (I) is an ideal of R.<br />
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2b. If P is a prime ideal of S, show that phi^(-1) (P) is either R or a prime ideal of R.<br />
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2c. If M is a maximal ideal of S and phi is surjective, show that phi^(-1) (M) is a maximal ideal of R.<br />
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2d. Find a homomorphism phi : R -> S and a maximal ideal M of S such that phi^(-1) (M) is not a maximal ideal of R.<br />
<br />
<br />
3. Let Nil(A) denote the set of nilpotent elements of a ring A, and let R be a commutative ring. In class it was observed that Nil(R) is an ideal of R.<br />
<br />
3a. Find (a generator of) Nil(Z/720Z).<br />
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3b. Show that Nil(R/Nil(R)) = 0.<br />
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3c. If x is in Nil(R) and u is a unit, show that u+x is a unit.<br />
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3d. Show that Nil(S), where S is the 2x2 matrices with real coefficients, is not an ideal of S.<br />
<br />
<br />
4a. If M is an R-module and I is an ideal of R such that im = 0 for all i in I and m in M, show that M carries the structure of an R/I-module.<br />
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4b. If S is a commutative ring with 1 and J is a maximal ideal of S, show that J/J^2 is a vector space over S/J.<br />
<br />
<br />
5. A "divisible element" x of a Z-module is one such that for each positive integer k, there exists a y such that ky=x. Show that a free Z-module cannot contain a nonzero divisible element.<br />
<br />
<br />
6. Prove that the direct sum of any collection of free R-modules is also a free R-module.<br />
<br />
<br />
7. In this problem you will prove that the module M given by the direct product of copies of Z, indexed by the positive integers, is not a free Z-module. Therefore, assume that M is a free Z-module with basis B, and let N be the direct sum of copies of Z, as a submodule of M.<br />
<br />
7a. Show there is a countable subset B1 of B such that N is contained in the submodule N1 generated by B1. [Hint: N is countable.]<br />
<br />
7b. Show that M/N1 is a free Z-module.<br />
<br />
7c. Let S = {(b1, b2, b3, ...) : b_i = i! or -i!}. Show that there is some s in S which is not in N1. [Hint: S is uncountable.]<br />
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7d. For any s in S but not N1, show that for each positive integer k, there exists some m in M with s-km in N1. Conclude that s+N1 is a divisible element of M/N1.<br />
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7e. Use problem 5 to obtain a contradiction.<br />
<br />
<br />
8. In class I said "rank" is not always well-defined for R-modules if R is not commutative. Here is an example demonstrating this assertion. Let M be the countably infinite direct product of copies of Z as in exercise 7, and let R be the endomorphism ring of M. Define phi_1(a1,a2,a3,a4,...) = (a1,a3,a5,...) and phi_2(a1,a2,a3,a4,...) = (a2,a4,a6,...).<br />
<br />
8a. Show that {phi_1, phi_2} is a free basis for R, as a left R-module. [Hint: let psi_1(a1,a2,a3,...) = (a1,0,a2,0,...) and psi_2(a1,a2,a3,...) = (0,a1,0,a2,...). Compute the products of the phi_i and psi_j in either order, and use the result to show that the phi_i are a basis.]<br />
<br />
8b. Show that R is isomorphic to R^n for every positive integer n.<br />
<br />
<br />
==HOMEWORK 10 (due Dec 4)==<br />
<br />
1. '''How algebraists do calculus'''. The ''ring of dual numbers'' is the ring R = C[e]/(e^2). We think of e as an infinitesimal number; that is, something so small that its square is 0. This algebraic notion of infinitesimal is the basis for the algebraic notion of differentiation, which works as follows. Let f be a polynomial in R[x]. Show that there is a unique polynomial g in C[x] such that<br />
<br />
f(x+e) - f(x) = eg(x)<br />
<br />
and that in fact g is the derivative of f. In other words, the usual definition of "derivative" works just fine in this context, without any use of the notion of limit!<br />
<br />
2. Write down all the ideals of C[e]/e^2. (There are three.)<br />
<br />
3. Show that any right ideal of the ring M_2(Q) of 2x2 matrices is an ideal of the form I_V described in class for some subspace V of Q^2; that is, it consists precisely of those linear transformations whose image is contained in V.<br />
<br />
<br />
4. An ''idempotent'' element of a ring is an element e satisfying e^2 = e.<br />
<br />
4a. If e is an idempotent, show that 1-e is also an idempotent.<br />
<br />
4b. Suppose R has a unit (we are actually always supposing this, but I just want to remind you of it for this problem) and suppose e is a ''central'' idempotent, i.e. an idempotent which commutes with all elements of R. Show that eR and (1-e)R are both two-sided ideals. Moreover, show that eR and (1-e)R are also subrings of R (with the warning that the identity in eR is not 1, which is not contained in eR, but rather e.)<br />
<br />
4c. Show that the map f_e: R -> eR defined by f_e(r) = er is a ring homomorphism. Finally show that the ring homomorphism<br />
<br />
f_{e} x f_{1-e}: R -> eR x (1-e)R <br />
<br />
is actually an isomorphism of rings. In other words, the presence of a central idempotent induces a decomposition a ring as a direct product.<br />
<br />
<br />
5. The ''center'' of a ring R is the subring of elements commuting with every element of R.<br />
<br />
5a. Show that the center of M_n(Q) is the ring of scalar matrices.<br />
<br />
5b. Show that the center of the group ring C[G] is the set of elements sum f(g) g such that f(g) is a class function.<br />
<br />
<br />
6. '''How algebraists do differential equations.''' We define the ''Weyl algebra'' W of differential operators in one variable as follows. Multiplication by x is a linear transformation on the complex vector space V of all smooth functions on C. Differentiation, which we denote d, is also a linear transformation on V. So we let W be the subring of End(V) generated by x and d. We note that x and d satisfy the relation xd = dx - 1; in particular, the Weyl algebra is not commutative.<br />
<br />
6a. Describe the submodule of V which is annihilated by the element d-x in W.<br />
<br />
6b. Show that, for any polynomial f(x), fd-df = -f', and also show that for any polynomial g(d), gx-xg = -g'. (This is another way algebraists do calculus!) <br />
<br />
6c. Use the facts from (b) to show that the center of the Weyl algebra is just the field of constants. (Remark: you will use in the course of this proof the fact that a polynomial whose derivative is 0 is constant -- this would not be true if we were working in positive characteristic, as the example of x^p considered as a polynomial over F_p demonstrates.)<br />
<br />
<br />
7. If M is a left module for a ring R, we denote by Ann(M) the set of elements r in R such that rm = 0 for all m in M. Show that Ann(M) is a two-sided ideal. <br />
<br />
<br />
8. Let M be a left module for R, and let m be an element of M. We denote by Ann(m) the set of elements r in R such that rm = 0.<br />
<br />
8a. Show that Ann(m) is a left ideal, and show that the submodule Rm of M generated by m is isomorphic to the left module R/Ann(m).<br />
<br />
8c. Let V,W be as in question 6. Show that Ann(e^x) is the left ideal W(d-1).<br />
<br />
==HOMEWORK 11 (due Dec 13)==<br />
<br />
1. Let f: R -> S be a homomorphism of rings. As discussed in class, if M is a left R-module, then S tensor_R M is a left S-module, and if N is an S-module, then N can also be considered as an R-module by pullback of the action.<br />
<br />
Prove that if M is a finitely generated left R-module, then S tensor_R M is a finitely generated left S-module. On the other hand, give an example to show that N can be finitely generated as a left S-module but not finitely generated when considered as a left R-module.<br />
<br />
2. Recall from class that when V and W are complex vector spaces, W^dual tensor V is naturally isomorphic to the space of linear homomorphisms Hom(W,V).<br />
<br />
Now suppose that V and W are left C[G]-modules, i.e. complex representations of a group G. Show that W^dual tensor_C[G} V is the maximal G-invariant quotient of Hom(W,V); in other words, the quotient of Hom(W,V) by the subspace generated by all functions F of the form F(w) = f(gw) - gf(w) for some function f in Hom(W,V) and some g in G.<br />
<br />
3. '''How algebraists do logic.''' Why are Boolean rings called Boolean rings? Because they model some features of the Boolean arithmetic that governs propositional logic. Let A be a collection of propositions. We define two operations on A as follows. By a + b we mean the negation of a xor b, while by ab we mean "a or b."<br />
<br />
3a. It is not a priori obvious that these operations satisfy the ring axioms, but assume this for the moment. Show that a^2 = a for all a (so A is a boolean ring) and 2a = 0 for all a (so A has characteristic 2.)<br />
<br />
3b. Suppose we have a consistent assignment of truth-values to the propositions in A. Show that the set of true statements of A forms an ideal of A.<br />
<br />
3c. (Optional for people who like propositional logic) Show that a(b+c) = ab + ac with the given operations.<br />
<br />
<br />
4. Let K be Q[x]/(x^3-2).<br />
<br />
4a. Show that K is a field.<br />
<br />
4b. Show that K tensor_Q K decomposes as the direct sum of two fields. What are they? [Note: K tensor_Q K is a ring!]<br />
<br />
5. There was some discussion in class about when we can "go back and forth" between right modules and left modules, as is the case with group rings.<br />
<br />
5a. If R is a ring, we denote R^op ("the opposite ring") to be the ring whose elements are those of R, but whose multiplication law x_opp is defined by s x_opp r = rs. Show that if M is a left R-module, then the map M x R^op -> M sending (x,r) to rx makes M into a right R^op module. In particular, if there is a RING ISOMORPHISM from R to R^op, then every left R-module can be thought of as a right R-module.<br />
<br />
5b. Verify that the map sending g to g^{-1} extends by linearity to give a ring isomorphism between C[G] and C[G]^op.<br />
<br />
5c. Construct a ring isomorphism between M_n(Q) and M_n(Q)^op.<br />
<br />
5d. Optional: Give an example of a ring (with 1) which is not isomorphic to its opposite ring. (This is not so easy, I think! I found a MathOverflow thread with several examples, none of which came with a short proof of the non-isomorphism -- try your hand!)<br />
<br />
6. . Let R be a commutative ring with 1. If R has a unique maximal ideal M, R is called a "local ring".<br />
<br />
6a. If R is a local ring, show that every element of R that is not in M is a unit.<br />
<br />
6b. Conversely, if R is such that the set of nonunits forms an ideal I, show that R is a local ring with maximal ideal I.<br />
<br />
7. The inclusion of S_3 into S_4 associated to the inclusion of {1,2,3} into {1,2,3,4} induces a ring homomorphism from C[S_3} to C{S_4]. Let V be a representation of C{S_3]. Then, as discussed in class, V tensor_C[S_3] C[S_4] is a representation of S_4, called the '''induced representation''' of V.<br />
<br />
7a. Show that dim_C (V tensor_C[S_3] C[S_4]) = 4 dim_C V.<br />
<br />
7b. When V is the trivial representation of S_3, decompose (V tensor_C[S_3] C[S_4]) as a sum of irreducible representations of S_4.<br />
<br />
--><br />
<br />
<!--==HOMEWORK PROBLEMS TO GIVE IN FUTURE SEMESTERS==<br />
<br />
Let f: C[x] tensor_C C[y] -> C[z] be the map of rings with f(x) = f(y) = z. Show that f(m tensor n) is not 0 for any pure tensor m tensor n, but f is not injective. (This is a warning not to try to check injectivity by checking that no pure tensors are killed by the map.)<br />
<br />
--></div>Dummithttps://hilbert.math.wisc.edu/wiki/index.php?title=741&diff=60107412013-09-29T04:08:48Z<p>Dummit: </p>
<hr />
<div>'''Math 741'''<br />
<br />
Algebra<br />
<br />
Prof: [http://www.math.wisc.edu/~ellenber Jordan Ellenberg]<br />
<br />
Grader: Evan Dummit. Homework policies:<br />
* Late homework may be given directly to the grader, along with either (i) the instructor's permission, or (ii) a polite request for mercy.<br />
* Assignments that are more than 1 page should be affixed in some reasonable way.<br />
* Results from places (e.g., the internet) other than 741 and standard books must be cited.<br />
<br />
JE's office hours: Monday 12pm-1pm (right after class)<br />
<br />
This course, the first semester of the introductory graduate sequence in algebra, will cover the basic theory of groups, group actions, representation, linear and multilinear algebra, and the beginnings of ring theory.<br />
<br />
==SYLLABUS==<br />
<br />
In this space we will record the theorems and definitions we covered each week, which we can use as a list of notions you should be prepared to answer questions about on the Algebra qualifying exam. The material covered on the homework is also an excellent guide to the scope of the course. <br />
<br />
'''WEEK 1''': <br />
<br />
Definition of group. Associativity. Inverse. <br />
<br />
Examples of group: GL_n(R). GL_n(Z). Z/nZ. R. Z. R^*. The free group F_k on k generators. <br />
<br />
Homomorphisms. The homomorphisms from F_k to G are in bijection with G^k. Isomorphisms.<br />
<br />
'''WEEK 2''':<br />
<br />
The symmetric group (or permutation group) S_n on n letters. Cycle decomposition of a permutation. Order of a permutation. Thm: every element of a finite group has finite order. <br />
<br />
Subgroups. Left and right cosets. Lagrange's Theorem. Cyclic groups. The order of an element of a finite group is a divisor of the order of the group.<br />
<br />
The sign homomorphism S_n -> +-1.<br />
<br />
<br />
'''WEEK 3'''<br />
<br />
Normal subgroups. The quotient of a group by a normal subgroup. The first isomorphism theorem. Examples of S_n -> +-1 and S_4 -> S_3 with kernel V_4, the Klein 4-group.<br />
<br />
Centralizers and centers. Abelian groups. The center of SL_n(R) is either 1 or +-1.<br />
<br />
Groups with presentations. The infinite dihedral group <x,y | x^2 = 1, y^2 = 1>.<br />
<br />
'''WEEK 4'''<br />
<br />
More on groups with presentations.<br />
<br />
Second and third isomorphism theorems.<br />
<br />
Semidirect products.<br />
<br />
'''WEEK 5'''<br />
<br />
Group actions, orbits, and stabilizers.<br />
<br />
Orbit-stabilizer theorem.<br />
<br />
Cayley's theorem.<br />
<br />
Cauchy's theorem.<br />
<br />
'''WEEK 6'''<br />
<br />
Applications of orbit-stabilizer theorem (p-groups have nontrivial center, first Sylow theorem.)<br />
<br />
Classification of finite abelian groups and finitely generated abelian groups.<br />
<br />
Composition series and the Jordan-Holder theorem (which we state but don't prove.) <br />
<br />
The difference between knowing the composition factors and knowing the group (e.g. all p-groups of the same order have the same composition factors.)<br />
<br />
'''WEEK 7'''<br />
<br />
Simplicity of A_n.<br />
<br />
Nilpotent groups (main example: the Heisenberg group)<br />
<br />
Derived series and lower central series.<br />
<br />
Category theory 101: Definition of category and functor. Some examples. A group is a groupoid with one object.<br />
<br />
'''WEEK 8'''<br />
<br />
Introduction to representation theory.<br />
<br />
'''WEEK 10'''<br />
<br />
Ring theory 101: Rings, ring homomorphisms, ideals, isomorphism theorems. Examples: fields, Z, the Hamilton quaternions, matrix rings, rings of polynomials and formal power series, quadratic integer rings, group rings. Integral domains. Maximal and prime ideals. The nilradical.<br />
<br />
Module theory 101: Modules, module homomorphisms, submodules, isomorphism theorems. Noetherian modules and Zorn's lemma. Direct sums and direct products of arbitrary collections of modules.<br />
<br />
<br />
==HOMEWORK 1 (due Sep 16)==<br />
<br />
1. Suppose that H_1 and H_2 are subgroups of a group G. Prove that the intersection of H_1 and H_2 is a subgroup of G.<br />
<br />
2. Recall that S_3 (the symmetric group) is the group of permutations of the set {1..3}. List all the subgroups of S_3.<br />
<br />
3. We can define an equivalence relation on rational numbers by declaring two rational numbers to be equal whenever they differ by an integer. We denote the set of equivalence classes by Q/Z. The operation of addition makes Q/Z into a group.<br />
<br />
a) For each n, prove that Q/Z has a subgroup of order n.<br />
<br />
b) Prove that Q/Z is a ''divisible'' group: that is, if x is an element of Q/Z and n is an integer, there exists an element y of Q/Z such that ny = x. (Note that we write the operation in this group as addition rather than multiplication, which is why we write ny for the n-fold product of y with itself rather than y^n)<br />
<br />
c) Prove that Q/Z is not finitely generated. (Hint: prove that if x_1, .. x_d is a finite subset of Q/Z, the subgroup of Q/Z generated by x_1, ... x_d is finite.)<br />
<br />
d) Conclude that Q is not finitely generated.<br />
<br />
4. We will prove that there is no homomorphism from SL_2(Z) to Z except the one which sends all of SL_2(Z) to 0. Suppose f is a homomorphism from SL_2(Z) to Z.<br />
<br />
a) Let U1 be the upper triangular matrix with 1's on the diagonal and a 1 in the upper right hand corner, as in class, and let U2 be the transpose of U1, also as in class. Show that (U1 U2^{-1})^6 = identity (JING 1, TAO 0) and explain why this implies that f(U1) = f(U2).<br />
<br />
b) Show that there is a matrix A in SL_2(Z) such that A U1 A^{-1} = U2^{-1}. (Recall that we say U1 and U2^{-1} are "conjugate".) Explain why this also implies that f(U1) = -f(U2).<br />
<br />
c) Explain why a) and b) imply that f must be identically 0.<br />
<br />
5. The argument above also shows that there is no nonzero homomorphism from SL_2(Z) to Z/pZ where p is a prime greater than 3. However, it leaves open the possibility that there is indeed a nonzero homomorphism from SL_2(Z) to Z/2Z. Exhibit such a homomorphism. Optional challenge problem: exhibit a nonzero homomorphism from SL_2(Z) to Z/3Z.<br />
<br />
<br />
==HOMEWORK 2 (due Sep 25)==<br />
<br />
1. Let V be the Klein 4-group in S_4. Let Q be the symmetric group on the set DF of double-flips in S_4; there are 3 double flips, so Q is isomorphic to S_3. If g is an element of S_4, we discussed in class that conjugation by g permutes the elements of DF. So to each element of g, we have associated an element f(g) of Q. More or less by definition, this defines a homomorphism f: G -> Q. Show that this homomorphism is surjective and has kernel H, and thus that G/H is isomorphic to Q.<br />
<br />
2. Let F_2 be the free group on two generators, which we denote x,y. Prove that there is a automorphism of F_2 which sends x to xyx and y to xy, and prove that this automorphism is unique.<br />
<br />
3. Let H be a subgroup of G, and let N_G(H), the ''normalizer'' of H in G, be the set of elements g in G satisfying g H g^{-1} = H. Prove that N_G(H) is a subgroup of G.<br />
<br />
4. Let T be the subgroup of GL_2(R) consisting of diagonal matrices. This is an example of a "Cartan subgroup," or "torus" (whence the notation T.) Describe the normalizer N(T) of T in GL_2(R). Prove that there is a homomorphism from N(T) to Z/2Z whose kernel is precisely T. <br />
<br />
5. Prove that the group of inner automorphisms of a group is normal in the full automorphism group Aut(G).<br />
<br />
6. Let H and H' be subgroups of a group G, and let x be an element of G. We denote by HxH' the set of elements of the form hxh', with h in H and h' in H', and we call HxH a ''double coset'' of the pair (H,H').<br />
<br />
a) Show that G decomposes as a disjoint union of double cosets of (H,H').<br />
<br />
b) Let G be GL_2(R) and let B be the subgroup of upper triangular matrices. Prove that G decomposes into exactly two double cosets of (B,B). (This is an example of the so-called *Bruhat decomposition* which is of great importance in the theory of algebraic groups and their representations.)<br />
<br />
c) Let G be the symmetric group S_n, and let H be the subgroup of permutations which fix 1. Describe the double coset decomposition of S_n into double cosets of (H,H).<br />
<br />
<br />
==HOMEWORK 3 (due Oct 2)==<br />
<br />
1. Compute the centralizer of the triple flip (12)(34)(56) in S_6. (Note: this is the kind of problem that a computer algebra package like sage is very good at, and I encourage you to learn to use sage well enough to check your answer.)<br />
<br />
2. Warning: a normal subgroup of a normal subgroup need not be normal! Let H be the subgroup of S_4 generated by (12)(34). Then H is a subgroup of the Klein 4-group V_4, which we have already shown is normal in S_4. Show that H is normal in V_4, but H is not normal in S_4.<br />
<br />
3. Let G be a group and let H_1 and H_2 be subgroups of G of finite index (that is, the number of cosets in G/H_i is finite, i = 1,2.) Prove that H_1 intersect H_2 also has finite index in G.<br />
<br />
4. Let X be the set of lines in F_3^2 (here a line means a 1-dimensional linear subspace) so that |X| = 4. Let G be the projective linear group PGL_2(F_3), so that G acts on X via its action on F_3^2. This action can be thought of as a homomorphism from G to S_4. Show that this homomorphism is an ISOMORPHISM from PGL_2(F_3) to S_4.<br />
<br />
5. We proved two theorems in class: Lagrange's theorem, which says that the order of a subgroup of G divides |G|, and Cauchy's theorem, which says that if p divides |G| then there exists a subgroup of G of order p. Cauchy's theorem is a kind of partial converse to Lagrange's theorem, but the full converse is false. Give an example of a finite group G and a divisor n of |G| such that you can prove there is no subgroup of G of order n.<br />
<br />
6. The group S_3 has only three conjugacy classes. Prove that a finite group with at most three conjugacy classes has order at most 6.<br />
<br />
7. Let X be the set of ordered triples of elements of {1,..,n}, for some n >= 3. Then S_n acts on X. How many orbits are there? (Hint: the answer does not depend on n.) OPTIONAL: How many orbits are there on the space of ordered k-tuples, when n >= k? <br />
<br />
<br />
<!--Below you will find a repository of homework problems. Note that some of these problems are taken from Lang's ''Algebra''.<br />
<br />
==HOMEWORK 1 (due Sep 20)==<br />
<br />
1. Suppose that H_1 and H_2 are subgroups of a group G. Prove that the intersection of H_1 and H_2 is a subgroup of G.<br />
<br />
2. Recall that S_3 (the symmetric group) is the group of permutations of the set {1..3}. List all the subgroups of S_3.<br />
<br />
3. We can define an equivalence relation on rational numbers by declaring two rational numbers to be equal whenever they differ by an integer. We denote the set of equivalence classes by Q/Z. The operation of addition makes Q/Z into a group.<br />
<br />
a) For each n, prove that Q/Z has a subgroup of order n.<br />
<br />
b) Prove that Q/Z is a ''divisible'' group: that is, if x is an element of Q/Z and n is an integer, there exists an element y of Q/Z such that ny = x. (Note that we write the operation in this group as addition rather than multiplication, which is why we write ny for the n-fold product of y with itself rather than y^n)<br />
<br />
c) Prove that Q/Z is not finitely generated. (Hint: prove that if x_1, .. x_d is a finite subset of Q/Z, the subgroup of Q/Z generated by x_1, ... x_d is finite.)<br />
<br />
4. Let F_2 be the free group on two generators x,y. Prove that there is a automorphism of F_2 which sends x to xyx and y to xy, and prove that this automorphism is unique.<br />
<br />
5. Let H be a subgroup of G, and let N_G(H), the ''normalizer'' of H in G, be the set of elements g in G satsifying g H g^{-1} = H. Prove that N_G(H) is a subgroup of G.<br />
<br />
6. Let T be the subgroup of GL_2(R) consisting of diagonal matrices. This is an example of a "Cartan subgroup," or "torus" (whence the notation T.) Describe the normalizer N(T) of T in GL_2(R). Prove that there is a homomorphism from N(T) to Z/2Z whose kernel is precisely T. <br />
<br />
7. Prove that the group of inner automorphisms of a group is normal in the full automorphism group Aut(G).<br />
<br />
8. Let H and H' be subgroups of a group G, and let x be an element of G. We denote by HxH' the set of elements of the form hxh', with h in H and h' in H', and we call HxH a ''double coset'' of the pair (H,H').<br />
<br />
a) Show that G decomposes as a disjoint union of double cosets of (H,H').<br />
<br />
b) Let G be GL_2(R) and let B be the subgroup of upper triangular matrices. Prove that G decomposes into exactly two double cosets of (B,B).<br />
<br />
c) Let G be the symmetric group S_n, and let H be the subgroup of permutations which fix 1. Describe the double coset decomposition of S_n into double cosets of (H,H).<br />
<br />
==HOMEWORK 2 (due Sep 27)==<br />
<br />
1. Prove that the symmetric group S_n is generated by transpositions. Hint: proceed by induction on n. Let H be the subgroup of S_n consisting of permutations fixing 1. Observe that H is isomorphic to S_{n-1}, so that H is generated by transpositions. Explain why it suffices to show that, for any g in S_n, the coset gH contains a transposition. Then explain why this is the case.<br />
<br />
2. Given that the symmetric group S_n is generated by transpositions, show that there are no nontrivial homomorphisms from S_n to Z/pZ for any odd prime p.<br />
<br />
3. Compute the centralizer of the triple flip (12)(34)(56) in S_6. (Note: this is the kind of problem that a computer algebra package like sage is very good at, and I encourage you to learn to use sage well enough to check your answer.)<br />
<br />
4. Warning: a normal subgroup of a normal subgroup need not be normal! Let H be the subgroup of S_4 generated by (12)(34). Then H is a subgroup of the Klein 4-group V_4, which we have already shown is normal in S_4. Show that H is normal in V_4, but H is not normal in S_4.<br />
<br />
5. The '''quaternion group''' Q is a non-abelian finite group of order 8. Its elements are 1, -1, i, -i, j, -j, k, -k; it satisfies g^2 = -1 for all g except +-1, and ijk = 1. (Note that the notation "-i" means "the product of i with -1" and -1 is understood to be central in Q.) Prove that every subgroup of Q is normal.<br />
<br />
6. (finishing example done in class) Let Gamma be the group F<x,y> / (x^2 = 1, y^2 = 1). (Remember, this means the quotient of the free group by the group generated by all conjugates of x^2 and y^2.) Prove that Gamma is infinite. Prove that Gamma has a subgroup R generated by xy isomorphic to Z, that R is normal, and that the quotient Gamma/R is isomorphic to Z/2Z. <br />
<br />
7. Suppose a group G has the property that every element in g satisfies g^2 = 1. Show that G must be abelian.<br />
<br />
==HOMEWORK 3 (due Oct 4)==<br />
<br />
1. Multiplication by -1 is an automorphism of Z/nZ. Let a be the homomorphism Z/2Z -> Aut(Z/nZ) which sends the nontrivial element of Z/2Z to multiplication by -1. Then we have a semidirect product of Z/pZ by Z/2Z as defined in class. This group of order 2n is called a ''dihedral group.'' and is denoted D_n (or sometimes D_{2n}).<br />
<br />
1a. Compute the center of D_n. (Note that the answer depends on n!)<br />
1b. Show that all involutions of D_n are conjugate to each other if and only if n is odd.<br />
<br />
2. (More on direct products that we didn't do in class.) Suppose that G is a semidirect product of N with H. Suppose furthermore that H is also normal in G. Prove that G is a direct product of N with H.<br />
<br />
3. Let Q be the quaternion group from last week's homework. Show that Q does not decompose as a semidirect product N \rtimes H, apart form the trivial cases N = {e} and N = Q.<br />
<br />
4. The ''affine linear group'' of degree n is the group of transformations from R^n to R^n of the form x -> Ax + b, where A is a matrix in GL_n(R) and b is a vector in R^n. Prove that the affine linear group is a semidirect product R^n \rtimes GL_n.<br />
<br />
5. The ''ordinary triangle group'' T(p,q,r) is the group with presentation <x,y | x^p = y^q = (xy)^r = 1>. This is a very interesting family of groups which arises naturally in many contexts in number theory and topology.<br />
<br />
6a. Prove that T(2,2,n) is isomorphic to the dihedral group D_n.<br />
6b. Prove that T(2,3,3) is finite and compute its order. Is it isomorphic to a finite group we've discussed in class?<br />
<br />
(It is a beautiful fact, but outside the scope of this course, that T(p,q,r) is finite if and only if 1/p + 1/q + 1/r > 1.)<br />
<br />
7. Let G be a subgroup of GL_n(R) which contains SL_n(R). Prove that G is a normal subgroup of GL_n(R).<br />
<br />
8. Let G be a group and let H_1 and H_2 be subgroups of G of finite index (that is, the number of cosets in G/H_i is finite, i = 1,2.) Prove that H_1 intersect H_2 also has finite index in G.<br />
<br />
9. Recall that the commutator [x,y] is x y x^{-1} y^{-1}. If G is a group, the commutator subgroup of G, denoted G' or [G,G], is the subgroup of G generated by all commutators [x,y] for x,y in G.<br />
<br />
9a. Show that G' is a normal subgroup of G.<br />
9b. Show that G/G' is an abelian group.<br />
9c. Show that if f: G -> A is a homomorphism, with A abelian, then G' is contained in ker f. Conclude that f factors through a homomorphism G/G' -> A.<br />
9d. Show that G' = G if and only if G has no nontrivial abelian quotient. In this case, we say G is ''perfect''.<br />
<br />
10. Under what conditions on (p,q,r) is the triangle group T(p,q,r) perfect?<br />
<br />
==HOMEWORK 4 (due Oct 16)==<br />
<br />
1. Let X be a set with a transitive action of a group G, which we think of as a homomorphism f: G -> Sym(X). Let H_x be the stabilizer of an element x of X.<br />
<br />
1a. If x' is another element of X, show that H_{x"} and H_x are conjugate subgroups. (Hint: this is not true without the hypothesis of transitivity, so make sure to use it!)<br />
<br />
1b. Show that the kernel of f is the intersection of all the conjugates of H_x in G. (This subgroup is called the ''normal core'' of H_x.)<br />
<br />
2. This week, a certified mathematical grown-up asked on MathOverflow what the index-3 subgroups of SL_2(Z/pZ) were. Let's do it ourselves for homework. Let p be a prime, and suppose H is an index-3 subgroup of SL_2(Z/pZ). I do not claim that the proof sketched below is necessarily the easiest, but it allows us to cover some ground that we won't get to do in lecture.<br />
<br />
2a. Using the coset action, show that the normal core of H is the kernel of a homomorphism f from SL_2(Z/pZ) to S_3.<br />
<br />
2b. A ''unipotent matrix'' is a matrix A such that (A-I)^n = 0 for some positive integer n. Prove that a unipotent element of SL_2(Z/pZ) has order dividing p.<br />
<br />
2c. You may use the fact that unipotent matrices generate SL_2(Z/pZ). Given this fact, show that f must be trivial when p > 3.<br />
<br />
2d. Using the above, show that SL_2(Z/pZ) has no index-3 subgroups when p > 3.<br />
<br />
2e. (extra credit) What can you say about the minimal index of a proper subgroup of SL_2(Z/pZ)?<br />
<br />
3. Let H be a subgroup of G of index 2. Prove that H is normal.<br />
<br />
4. We proved two theorems in class: Lagrange's theorem, which says that the order of a subgroup of G divides |G|, and Cauchy's theorem, which says that if p divides |G| then there exists a subgroup of G of order p. Cauchy's theorem is a kind of partial converse to Lagrange's theorem, but the full converse is false. Give an example of a finite group G and a divisor n of |G| such that you can prove there is no subgroup of G of order n.<br />
<br />
5. Let X be the set of lines in F_3^2 (here a line means a 1-dimensional linear subspace) so that |X| = 4. Let G be the projective linear group PGL_2(F_3), so that G acts on X via its action on F_3^2. This action can be thought of as a homomorphism from G to S_4. Show that this homomorphism is an ISOMORPHISM from PGL_2(F_3) to S_4.<br />
<br />
6. The group S_3 has only three conjugacy classes. Prove that a finite group with at most three conjugacy classes has order at most 6.<br />
<br />
7. Let X be the set of ordered triples of elements of {1,..,n}, for some n >= 3. Then S_n acts on X. How many orbits are there? (Hint: the answer does not depend on n.)<br />
<br />
8. Suppose that G and H are groups and f: G -> H is a homomorphism. Recall the definition of the abelianization G^ab from the previous problem set.<br />
<br />
8a. Show that the composition G -> H -> H^ab factors through a unique homomorphism G^ab -> H^ab, which we denote f^ab.<br />
<br />
8b. Show that if f: G -> H and g: H -> Q are homomorphisms, then f^ab o g^ab = (f o g)^ab.<br />
<br />
(For those reading MacLane, this constitutes a proof that abelianization is a '''functor''' from the category of groups to the category of abelian groups.)<br />
<br />
==HOMEWORK 5 (due Oct 23)==<br />
<br />
1. Show that there are only two different isomorphism classes of groups of order 14. (Hint: prove that such a group must be a semidirect product of a group of order 7 by a group of order 2.)<br />
<br />
2. (variant of Burnside's Lemma) Let G be a finite group and H a normal subgroup. Suppose G acts on a finite set X. <br />
<br />
2a. Show that if x and y lie in the same H-orbit, and g is an element of G, then gx and gy also lie in the same H-orbit. Thus, G/H has a natural action on the set of H-orbits X/H.<br />
<br />
2b. Show that the average number of fixed points in X of an element of the coset gH is the same as the number of fixed points of the element gH of G/H in its action on X/H.<br />
<br />
3. A ''central extension'' of a group G by an abelian group A is a group E, together with a surjective homomorphism E -> G whose kernel is central in E and is isomorphic to A.<br />
<br />
3a. Show that the quaternion group is a central extension of (Z/2Z)^2 by Z/2Z.<br />
<br />
3b. Give an example of another nonabelian group which is a central extension of (Z/2Z)^2 by Z/2Z and which is not isomorphic to the quaternion group.<br />
<br />
4. Prove that a central extension of an abelian group is nilpotent.<br />
<br />
5. Give two different composition series for S_4 and show that they have the same composition factors.<br />
<br />
6. Suppose that A is a finitely generated abelian group, and B is a subgroup of A.<br />
<br />
6a. Prove that B is finitely generated and that rank(B) <= rank(A).<br />
<br />
6b. Prove that rank(A/B) = rank(A) - rank(B).<br />
<br />
7. For this problem, you may use the fact (which I should have asked on homework earlier) that if H_1 and H_2 are subgroups of G, then<br />
<br />
[G: H_1 intersect H_2] <= [G:H_1][G:H_2].<br />
<br />
Prove that if G is a simple group, d is an integer greater than 1, and S is a subgroup of index d in G, then |G| <= d^d.<br />
<br />
Conclude that, in particular, A_5 has no subgroup of index 3. On the other hand, exhibit a subgroup of A_5 of index 5.<br />
<br />
(Note: d^d is not a sharp bound; can you get a bound of d factorial instead?)<br />
<br />
<!--(This argument shows that a solvable subgroup of A_n has size at most about (n-1)!, but this is far from sharp; e.g. I found a citation to a 1967 paper of Dixon which shows that a solvable subgroup of A_n has size at most k^n where k is around 2.88.)--><br />
<br />
<!--<br />
8. Give an example of an abelian group A with a subgroup B such that B is not a direct summand of A.<br />
<br />
9. Let A be the subgroup of Z^2 generated by the vector (1,0). Let B be the subgroup of Z^2 generated by the vector (3,3). Show that A and B are isomorphic to each other, but Z^2 / A is not isomorphic to Z^2 / B.<br />
<br />
==HOMEWORK 6 (due Oct 30)==<br />
<br />
1. In this problem, you will construct a functor F from the category of finite sets to the category of complex vector spaces. I'll tell you what it does on objects: if X is a finite set, then F(X) is the vector space spanned by a set of basis elements {e_x}_{x in X}, so that dim F(X) = |X|. Your job: complete the definition of F by describing the map from Mor_{Finite Sets}(X,Y) to Mor_{Vector Spaces}(F(X),F(Y)), and showing that your map respects composition.<br />
<br />
2. If (V, rho) is a representation of a group G, denote by V^G the space of vectors in V such that gv = v for all g in G; these are called _invariant_ vectors. We saw that the permutation representation of S_3 has a one-dimensional space of invariants. Prove that if X is a finite set with G-action, and V_X the corresponding permutation representation of G, then<br />
<br />
dim V_X^G = number of orbits of X.<br />
<br />
3. (Invariants are functorial) Let G be a group. Then there is a category called C[G]-Mod whose objects are complex representations of G (i.e. pairs (V,rho) where V is a complex vector space and rho is a homomorphism from G to GL(V)). The morphisms in this category are the ones described in class: Mor((V,rho),(W,psi)) is the set of linear maps f from V to W such that<br />
<br />
f(rho(g)(v)) = psi(g)(f(v))<br />
<br />
for all g in G and all v in V.<br />
<br />
Show that there is a functor H_0 from the category of representations of G to the category of vector spaces whose action on objects is given by<br />
<br />
H_0((V) = V^G.<br />
<br />
(In other words: your job again is to explain how to associate to a morphism from (V,rho) to (W,phi) a morphism from V^G to W^G, in a way that's compatible with composition.)<br />
<br />
<!--4. The group PSL_2(Z) has a presentation of the form [U,V|U^2 = V^3 = 1]. Thus, an n-dimensional representation of PSL_2(Z) is precisely a pair of n x n matrices (rho(U),rho(V)), such that rho(U)^2 = rho(V)^3 = 1. Prove that for every n > 1 there are uncountably many isomorphism classes of n-dimensional representations of PSL_2(Z). (Extra credit: show that there are uncountably many isomorphism classes of _irreducible_ representations.)<br />
<br />
An amazing theorem of Margulis (superrigidity) shows that SL_2(Z) is very special among arithmetic groups in this way. For instance, there are only _finitely_ many isomorphism classes of n-dimensional representations of PSL_d(Z) for any d > 2; those groups are "rigid."--><br />
<br />
<!--<br />
5. Let G be a finite group, let X be a set with G-action, and let chi be the character of the corresponding permutation representation. Show that, for each g in G, chi(g) is the number of elements of X fixed by g. (I am actually going to prove this in class.)<br />
<br />
6. Let V be the space of homogeneous degree-3 polynomials in variables x_1, x_2, x_3; then S_3 acts on V by permutation of the three variables. Describe the decomposition of V into irreducible representations of S_3.<br />
<br />
7a. Show that if chi is the character of a representation of a finite group G, then chi(g^{-1}) is the complex conjugate of chi.<br />
<br />
7b. As a consequence, show that if g is conjugate to g^{-1} for all g in G, then chi(g) is real.<br />
<br />
7c. The groups D_p and S_n have the property that g is conjugate to g^{-1} for _every_ g in G, so that every character is a real-valued function on G. Give an example of a group G where this is not the case, and give an example of a representation of G whose character takes non-real values.<br />
<br />
==HOMEWORK 7 (due Nov 6)==<br />
<br />
1. Let G be a finite group, let X be a finite set with G-action, and let chi be the character of the corresponding permutation representation. Show that, for each g in G, chi(g) is the number of elements of X fixed by g. (I am actually going to prove this in class.)<br />
<br />
2a. Show that if chi is the character of a finite-dimensional representation of a finite group G, then chi(g^{-1}) is the complex conjugate of chi(g) for all g.<br />
<br />
2b. As a consequence, show that if g is conjugate to g^{-1} for all g in G, then chi(g) is real.<br />
<br />
2c. The groups D_p and S_n have the property that g is conjugate to g^{-1} for _every_ g in G, so that every character is a real-valued function on G. Give an example of a group G where this is not the case, and give an example of a representation of G whose character takes non-real values.<br />
<br />
3. Let V and W be complex vector spaces. Let f be a linear transformation of V, and g a linear transformation of W. <br />
<br />
3a. Show that there is a unique linear transformation F satisfying<br />
<br />
F(v tensor w) = f(v) tensor g(w)<br />
<br />
for all v in V and all w in W. We denote this transformation by f tensor g.<br />
<br />
3b. Suppose V and W are finite dimensional. Show that Trace(f tensor g) = Trace(f) Trace(g). What does this say when f and g are both the identity transformation?<br />
<br />
4. Let V be a vector space. We define Sym^2 V to be the quotient of V tensor V by the subspace generated by all elements of the form<br />
<br />
v tensor w - w tensor v<br />
<br />
for v,w in V.<br />
<br />
Suppose dim V = n. What is dim Sym^2 V?<br />
<br />
5. Show that Z/pZ tensor Z/qZ is zero when p and q are distinct primes.<br />
<br />
==HOMEWORK 8 (due Nov 13)==<br />
<br />
1. (Schur's lemma can be false over non-algebraically closed fields) Let G be Z/3Z and let g be a generator of G. Then G acts by cyclic permutations on the set {1,2,3}; let P be the corresponding permutation representation on the REAL (not complex!) vector space generated by e_1,e_2,e_3.<br />
<br />
1a. Show that P breaks up as the direct sum of the trivial representation and a 2-dimensional representation V which is irreducible. (Remember, we are not working over the complex numbers, so you cannot use the criterion for irreducibility in terms of the character that we worked out in class for the complex case; you need to prove directly that V has no nontrivial proper subrepresentation.)<br />
<br />
1b. Show that the map from V to V induced by g is G-equivariant, but is neither zero nor a scalar multiplication.<br />
<br />
2. If V_1 and V_2 are representations (not necessarily irreducible!) of G, show that <chi_V_1, chi V_2> is a non-negative integer.<br />
<br />
3. When a formula always yields a non-negative integer, it is often because it is secretly a formula for either the cardinality of some finite set or the dimension of some finite-dimensional vector space. In this case, show that<br />
<br />
<chi_{V_1}, chi_{V_2}> = dim_C Hom_G(V_1,V_2)<br />
<br />
where we recall that Hom_G(V_1,G_2) refers to the space of G-equivariant homomorphisms from V_1 to V_2.<br />
<br />
4. If A is an abelian group, show that every irreducible representation of A is 1-dimensional. (Hint: show that there is a vector which is an eigenvector for every element of A!)<br />
<br />
5. Let G be the group of order 20 which is the semidirect product of A by H, where A is Z/5Z and H is (Z/5Z)^* with its natural action on A. We will work out all the irreducible representations of A. <br />
<br />
5a. First of all, consider the 5-dimensional permutation representation of G acting by left multiplication on the cosets of H. Show that this representation decomposes as a direct sum of the trivial representation and an irreducible 4-dimensional representation. (Hint: it is probably easier to use the character of this representation than to prove irreducibility directly.) <br />
<br />
5b. Now show that there are 4 1-dimensional representations of G which factor through the quotient G/A (i.e. such that rho(A) is the identity.)<br />
<br />
5c. Using the decomposition of the regular representation, show that the irreducible representations of G you have constructed are the only ones.<br />
<br />
==HOMEWORK 9 (due Nov 20)==<br />
<br />
0. Find all ring homomorphisms from Z to Z/30Z. [Warning: there are 8 of them, not 30.]<br />
<br />
<br />
1. If R is a ring (not necessarily with 1) such that a^2 = a for every a in R, R is called a Boolean ring.<br />
<br />
1a. Show that any nonzero Boolean ring has characteristic 2. (In other words, show that a+a=0 for all a in R.)<br />
<br />
1b. Show that every Boolean ring is commutative.<br />
<br />
<br />
2. Let phi : R -> S be a homomorphism of commutative rings with 1.<br />
<br />
2a. If I is an ideal of S, show that phi^(-1) (I) is an ideal of R.<br />
<br />
2b. If P is a prime ideal of S, show that phi^(-1) (P) is either R or a prime ideal of R.<br />
<br />
2c. If M is a maximal ideal of S and phi is surjective, show that phi^(-1) (M) is a maximal ideal of R.<br />
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2d. Find a homomorphism phi : R -> S and a maximal ideal M of S such that phi^(-1) (M) is not a maximal ideal of R.<br />
<br />
<br />
3. Let Nil(A) denote the set of nilpotent elements of a ring A, and let R be a commutative ring. In class it was observed that Nil(R) is an ideal of R.<br />
<br />
3a. Find (a generator of) Nil(Z/720Z).<br />
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3b. Show that Nil(R/Nil(R)) = 0.<br />
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3c. If x is in Nil(R) and u is a unit, show that u+x is a unit.<br />
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3d. Show that Nil(S), where S is the 2x2 matrices with real coefficients, is not an ideal of S.<br />
<br />
<br />
4a. If M is an R-module and I is an ideal of R such that im = 0 for all i in I and m in M, show that M carries the structure of an R/I-module.<br />
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4b. If S is a commutative ring with 1 and J is a maximal ideal of S, show that J/J^2 is a vector space over S/J.<br />
<br />
<br />
5. A "divisible element" x of a Z-module is one such that for each positive integer k, there exists a y such that ky=x. Show that a free Z-module cannot contain a nonzero divisible element.<br />
<br />
<br />
6. Prove that the direct sum of any collection of free R-modules is also a free R-module.<br />
<br />
<br />
7. In this problem you will prove that the module M given by the direct product of copies of Z, indexed by the positive integers, is not a free Z-module. Therefore, assume that M is a free Z-module with basis B, and let N be the direct sum of copies of Z, as a submodule of M.<br />
<br />
7a. Show there is a countable subset B1 of B such that N is contained in the submodule N1 generated by B1. [Hint: N is countable.]<br />
<br />
7b. Show that M/N1 is a free Z-module.<br />
<br />
7c. Let S = {(b1, b2, b3, ...) : b_i = i! or -i!}. Show that there is some s in S which is not in N1. [Hint: S is uncountable.]<br />
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7d. For any s in S but not N1, show that for each positive integer k, there exists some m in M with s-km in N1. Conclude that s+N1 is a divisible element of M/N1.<br />
<br />
7e. Use problem 5 to obtain a contradiction.<br />
<br />
<br />
8. In class I said "rank" is not always well-defined for R-modules if R is not commutative. Here is an example demonstrating this assertion. Let M be the countably infinite direct product of copies of Z as in exercise 7, and let R be the endomorphism ring of M. Define phi_1(a1,a2,a3,a4,...) = (a1,a3,a5,...) and phi_2(a1,a2,a3,a4,...) = (a2,a4,a6,...).<br />
<br />
8a. Show that {phi_1, phi_2} is a free basis for R, as a left R-module. [Hint: let psi_1(a1,a2,a3,...) = (a1,0,a2,0,...) and psi_2(a1,a2,a3,...) = (0,a1,0,a2,...). Compute the products of the phi_i and psi_j in either order, and use the result to show that the phi_i are a basis.]<br />
<br />
8b. Show that R is isomorphic to R^n for every positive integer n.<br />
<br />
<br />
==HOMEWORK 10 (due Dec 4)==<br />
<br />
1. '''How algebraists do calculus'''. The ''ring of dual numbers'' is the ring R = C[e]/(e^2). We think of e as an infinitesimal number; that is, something so small that its square is 0. This algebraic notion of infinitesimal is the basis for the algebraic notion of differentiation, which works as follows. Let f be a polynomial in R[x]. Show that there is a unique polynomial g in C[x] such that<br />
<br />
f(x+e) - f(x) = eg(x)<br />
<br />
and that in fact g is the derivative of f. In other words, the usual definition of "derivative" works just fine in this context, without any use of the notion of limit!<br />
<br />
2. Write down all the ideals of C[e]/e^2. (There are three.)<br />
<br />
3. Show that any right ideal of the ring M_2(Q) of 2x2 matrices is an ideal of the form I_V described in class for some subspace V of Q^2; that is, it consists precisely of those linear transformations whose image is contained in V.<br />
<br />
<br />
4. An ''idempotent'' element of a ring is an element e satisfying e^2 = e.<br />
<br />
4a. If e is an idempotent, show that 1-e is also an idempotent.<br />
<br />
4b. Suppose R has a unit (we are actually always supposing this, but I just want to remind you of it for this problem) and suppose e is a ''central'' idempotent, i.e. an idempotent which commutes with all elements of R. Show that eR and (1-e)R are both two-sided ideals. Moreover, show that eR and (1-e)R are also subrings of R (with the warning that the identity in eR is not 1, which is not contained in eR, but rather e.)<br />
<br />
4c. Show that the map f_e: R -> eR defined by f_e(r) = er is a ring homomorphism. Finally show that the ring homomorphism<br />
<br />
f_{e} x f_{1-e}: R -> eR x (1-e)R <br />
<br />
is actually an isomorphism of rings. In other words, the presence of a central idempotent induces a decomposition a ring as a direct product.<br />
<br />
<br />
5. The ''center'' of a ring R is the subring of elements commuting with every element of R.<br />
<br />
5a. Show that the center of M_n(Q) is the ring of scalar matrices.<br />
<br />
5b. Show that the center of the group ring C[G] is the set of elements sum f(g) g such that f(g) is a class function.<br />
<br />
<br />
6. '''How algebraists do differential equations.''' We define the ''Weyl algebra'' W of differential operators in one variable as follows. Multiplication by x is a linear transformation on the complex vector space V of all smooth functions on C. Differentiation, which we denote d, is also a linear transformation on V. So we let W be the subring of End(V) generated by x and d. We note that x and d satisfy the relation xd = dx - 1; in particular, the Weyl algebra is not commutative.<br />
<br />
6a. Describe the submodule of V which is annihilated by the element d-x in W.<br />
<br />
6b. Show that, for any polynomial f(x), fd-df = -f', and also show that for any polynomial g(d), gx-xg = -g'. (This is another way algebraists do calculus!) <br />
<br />
6c. Use the facts from (b) to show that the center of the Weyl algebra is just the field of constants. (Remark: you will use in the course of this proof the fact that a polynomial whose derivative is 0 is constant -- this would not be true if we were working in positive characteristic, as the example of x^p considered as a polynomial over F_p demonstrates.)<br />
<br />
<br />
7. If M is a left module for a ring R, we denote by Ann(M) the set of elements r in R such that rm = 0 for all m in M. Show that Ann(M) is a two-sided ideal. <br />
<br />
<br />
8. Let M be a left module for R, and let m be an element of M. We denote by Ann(m) the set of elements r in R such that rm = 0.<br />
<br />
8a. Show that Ann(m) is a left ideal, and show that the submodule Rm of M generated by m is isomorphic to the left module R/Ann(m).<br />
<br />
8c. Let V,W be as in question 6. Show that Ann(e^x) is the left ideal W(d-1).<br />
<br />
==HOMEWORK 11 (due Dec 13)==<br />
<br />
1. Let f: R -> S be a homomorphism of rings. As discussed in class, if M is a left R-module, then S tensor_R M is a left S-module, and if N is an S-module, then N can also be considered as an R-module by pullback of the action.<br />
<br />
Prove that if M is a finitely generated left R-module, then S tensor_R M is a finitely generated left S-module. On the other hand, give an example to show that N can be finitely generated as a left S-module but not finitely generated when considered as a left R-module.<br />
<br />
2. Recall from class that when V and W are complex vector spaces, W^dual tensor V is naturally isomorphic to the space of linear homomorphisms Hom(W,V).<br />
<br />
Now suppose that V and W are left C[G]-modules, i.e. complex representations of a group G. Show that W^dual tensor_C[G} V is the maximal G-invariant quotient of Hom(W,V); in other words, the quotient of Hom(W,V) by the subspace generated by all functions F of the form F(w) = f(gw) - gf(w) for some function f in Hom(W,V) and some g in G.<br />
<br />
3. '''How algebraists do logic.''' Why are Boolean rings called Boolean rings? Because they model some features of the Boolean arithmetic that governs propositional logic. Let A be a collection of propositions. We define two operations on A as follows. By a + b we mean the negation of a xor b, while by ab we mean "a or b."<br />
<br />
3a. It is not a priori obvious that these operations satisfy the ring axioms, but assume this for the moment. Show that a^2 = a for all a (so A is a boolean ring) and 2a = 0 for all a (so A has characteristic 2.)<br />
<br />
3b. Suppose we have a consistent assignment of truth-values to the propositions in A. Show that the set of true statements of A forms an ideal of A.<br />
<br />
3c. (Optional for people who like propositional logic) Show that a(b+c) = ab + ac with the given operations.<br />
<br />
<br />
4. Let K be Q[x]/(x^3-2).<br />
<br />
4a. Show that K is a field.<br />
<br />
4b. Show that K tensor_Q K decomposes as the direct sum of two fields. What are they? [Note: K tensor_Q K is a ring!]<br />
<br />
5. There was some discussion in class about when we can "go back and forth" between right modules and left modules, as is the case with group rings.<br />
<br />
5a. If R is a ring, we denote R^op ("the opposite ring") to be the ring whose elements are those of R, but whose multiplication law x_opp is defined by s x_opp r = rs. Show that if M is a left R-module, then the map M x R^op -> M sending (x,r) to rx makes M into a right R^op module. In particular, if there is a RING ISOMORPHISM from R to R^op, then every left R-module can be thought of as a right R-module.<br />
<br />
5b. Verify that the map sending g to g^{-1} extends by linearity to give a ring isomorphism between C[G] and C[G]^op.<br />
<br />
5c. Construct a ring isomorphism between M_n(Q) and M_n(Q)^op.<br />
<br />
5d. Optional: Give an example of a ring (with 1) which is not isomorphic to its opposite ring. (This is not so easy, I think! I found a MathOverflow thread with several examples, none of which came with a short proof of the non-isomorphism -- try your hand!)<br />
<br />
6. . Let R be a commutative ring with 1. If R has a unique maximal ideal M, R is called a "local ring".<br />
<br />
6a. If R is a local ring, show that every element of R that is not in M is a unit.<br />
<br />
6b. Conversely, if R is such that the set of nonunits forms an ideal I, show that R is a local ring with maximal ideal I.<br />
<br />
7. The inclusion of S_3 into S_4 associated to the inclusion of {1,2,3} into {1,2,3,4} induces a ring homomorphism from C[S_3} to C{S_4]. Let V be a representation of C{S_3]. Then, as discussed in class, V tensor_C[S_3] C[S_4] is a representation of S_4, called the '''induced representation''' of V.<br />
<br />
7a. Show that dim_C (V tensor_C[S_3] C[S_4]) = 4 dim_C V.<br />
<br />
7b. When V is the trivial representation of S_3, decompose (V tensor_C[S_3] C[S_4]) as a sum of irreducible representations of S_4.<br />
<br />
--><br />
<br />
<!--==HOMEWORK PROBLEMS TO GIVE IN FUTURE SEMESTERS==<br />
<br />
Let f: C[x] tensor_C C[y] -> C[z] be the map of rings with f(x) = f(y) = z. Show that f(m tensor n) is not 0 for any pure tensor m tensor n, but f is not injective. (This is a warning not to try to check injectivity by checking that no pure tensors are killed by the map.)<br />
<br />
--></div>Dummithttps://hilbert.math.wisc.edu/wiki/index.php?title=741&diff=60097412013-09-29T04:07:23Z<p>Dummit: </p>
<hr />
<div>'''Math 741'''<br />
<br />
Algebra<br />
<br />
Prof: [http://www.math.wisc.edu/~ellenber Jordan Ellenberg]<br />
<br />
Grader: Evan Dummit. Homework policies:<br />
* Late homework may be given directly to the grader, along with either (i) the instructor's permission, or (ii) a polite request for mercy.<br />
* Assignments that are more than 1 page should be stapled/paperclipped/affixed in some reasonable way.<br />
* If results from places other than this course and standard textbooks (e.g., from the internet) are used, they must be cited.<br />
<br />
JE's office hours: Monday 12pm-1pm (right after class)<br />
<br />
This course, the first semester of the introductory graduate sequence in algebra, will cover the basic theory of groups, group actions, representation, linear and multilinear algebra, and the beginnings of ring theory.<br />
<br />
==SYLLABUS==<br />
<br />
In this space we will record the theorems and definitions we covered each week, which we can use as a list of notions you should be prepared to answer questions about on the Algebra qualifying exam. The material covered on the homework is also an excellent guide to the scope of the course. <br />
<br />
'''WEEK 1''': <br />
<br />
Definition of group. Associativity. Inverse. <br />
<br />
Examples of group: GL_n(R). GL_n(Z). Z/nZ. R. Z. R^*. The free group F_k on k generators. <br />
<br />
Homomorphisms. The homomorphisms from F_k to G are in bijection with G^k. Isomorphisms.<br />
<br />
'''WEEK 2''':<br />
<br />
The symmetric group (or permutation group) S_n on n letters. Cycle decomposition of a permutation. Order of a permutation. Thm: every element of a finite group has finite order. <br />
<br />
Subgroups. Left and right cosets. Lagrange's Theorem. Cyclic groups. The order of an element of a finite group is a divisor of the order of the group.<br />
<br />
The sign homomorphism S_n -> +-1.<br />
<br />
<br />
'''WEEK 3'''<br />
<br />
Normal subgroups. The quotient of a group by a normal subgroup. The first isomorphism theorem. Examples of S_n -> +-1 and S_4 -> S_3 with kernel V_4, the Klein 4-group.<br />
<br />
Centralizers and centers. Abelian groups. The center of SL_n(R) is either 1 or +-1.<br />
<br />
Groups with presentations. The infinite dihedral group <x,y | x^2 = 1, y^2 = 1>.<br />
<br />
'''WEEK 4'''<br />
<br />
More on groups with presentations.<br />
<br />
Second and third isomorphism theorems.<br />
<br />
Semidirect products.<br />
<br />
'''WEEK 5'''<br />
<br />
Group actions, orbits, and stabilizers.<br />
<br />
Orbit-stabilizer theorem.<br />
<br />
Cayley's theorem.<br />
<br />
Cauchy's theorem.<br />
<br />
'''WEEK 6'''<br />
<br />
Applications of orbit-stabilizer theorem (p-groups have nontrivial center, first Sylow theorem.)<br />
<br />
Classification of finite abelian groups and finitely generated abelian groups.<br />
<br />
Composition series and the Jordan-Holder theorem (which we state but don't prove.) <br />
<br />
The difference between knowing the composition factors and knowing the group (e.g. all p-groups of the same order have the same composition factors.)<br />
<br />
'''WEEK 7'''<br />
<br />
Simplicity of A_n.<br />
<br />
Nilpotent groups (main example: the Heisenberg group)<br />
<br />
Derived series and lower central series.<br />
<br />
Category theory 101: Definition of category and functor. Some examples. A group is a groupoid with one object.<br />
<br />
'''WEEK 8'''<br />
<br />
Introduction to representation theory.<br />
<br />
'''WEEK 10'''<br />
<br />
Ring theory 101: Rings, ring homomorphisms, ideals, isomorphism theorems. Examples: fields, Z, the Hamilton quaternions, matrix rings, rings of polynomials and formal power series, quadratic integer rings, group rings. Integral domains. Maximal and prime ideals. The nilradical.<br />
<br />
Module theory 101: Modules, module homomorphisms, submodules, isomorphism theorems. Noetherian modules and Zorn's lemma. Direct sums and direct products of arbitrary collections of modules.<br />
<br />
<br />
==HOMEWORK 1 (due Sep 16)==<br />
<br />
1. Suppose that H_1 and H_2 are subgroups of a group G. Prove that the intersection of H_1 and H_2 is a subgroup of G.<br />
<br />
2. Recall that S_3 (the symmetric group) is the group of permutations of the set {1..3}. List all the subgroups of S_3.<br />
<br />
3. We can define an equivalence relation on rational numbers by declaring two rational numbers to be equal whenever they differ by an integer. We denote the set of equivalence classes by Q/Z. The operation of addition makes Q/Z into a group.<br />
<br />
a) For each n, prove that Q/Z has a subgroup of order n.<br />
<br />
b) Prove that Q/Z is a ''divisible'' group: that is, if x is an element of Q/Z and n is an integer, there exists an element y of Q/Z such that ny = x. (Note that we write the operation in this group as addition rather than multiplication, which is why we write ny for the n-fold product of y with itself rather than y^n)<br />
<br />
c) Prove that Q/Z is not finitely generated. (Hint: prove that if x_1, .. x_d is a finite subset of Q/Z, the subgroup of Q/Z generated by x_1, ... x_d is finite.)<br />
<br />
d) Conclude that Q is not finitely generated.<br />
<br />
4. We will prove that there is no homomorphism from SL_2(Z) to Z except the one which sends all of SL_2(Z) to 0. Suppose f is a homomorphism from SL_2(Z) to Z.<br />
<br />
a) Let U1 be the upper triangular matrix with 1's on the diagonal and a 1 in the upper right hand corner, as in class, and let U2 be the transpose of U1, also as in class. Show that (U1 U2^{-1})^6 = identity (JING 1, TAO 0) and explain why this implies that f(U1) = f(U2).<br />
<br />
b) Show that there is a matrix A in SL_2(Z) such that A U1 A^{-1} = U2^{-1}. (Recall that we say U1 and U2^{-1} are "conjugate".) Explain why this also implies that f(U1) = -f(U2).<br />
<br />
c) Explain why a) and b) imply that f must be identically 0.<br />
<br />
5. The argument above also shows that there is no nonzero homomorphism from SL_2(Z) to Z/pZ where p is a prime greater than 3. However, it leaves open the possibility that there is indeed a nonzero homomorphism from SL_2(Z) to Z/2Z. Exhibit such a homomorphism. Optional challenge problem: exhibit a nonzero homomorphism from SL_2(Z) to Z/3Z.<br />
<br />
<br />
==HOMEWORK 2 (due Sep 25)==<br />
<br />
1. Let V be the Klein 4-group in S_4. Let Q be the symmetric group on the set DF of double-flips in S_4; there are 3 double flips, so Q is isomorphic to S_3. If g is an element of S_4, we discussed in class that conjugation by g permutes the elements of DF. So to each element of g, we have associated an element f(g) of Q. More or less by definition, this defines a homomorphism f: G -> Q. Show that this homomorphism is surjective and has kernel H, and thus that G/H is isomorphic to Q.<br />
<br />
2. Let F_2 be the free group on two generators, which we denote x,y. Prove that there is a automorphism of F_2 which sends x to xyx and y to xy, and prove that this automorphism is unique.<br />
<br />
3. Let H be a subgroup of G, and let N_G(H), the ''normalizer'' of H in G, be the set of elements g in G satisfying g H g^{-1} = H. Prove that N_G(H) is a subgroup of G.<br />
<br />
4. Let T be the subgroup of GL_2(R) consisting of diagonal matrices. This is an example of a "Cartan subgroup," or "torus" (whence the notation T.) Describe the normalizer N(T) of T in GL_2(R). Prove that there is a homomorphism from N(T) to Z/2Z whose kernel is precisely T. <br />
<br />
5. Prove that the group of inner automorphisms of a group is normal in the full automorphism group Aut(G).<br />
<br />
6. Let H and H' be subgroups of a group G, and let x be an element of G. We denote by HxH' the set of elements of the form hxh', with h in H and h' in H', and we call HxH a ''double coset'' of the pair (H,H').<br />
<br />
a) Show that G decomposes as a disjoint union of double cosets of (H,H').<br />
<br />
b) Let G be GL_2(R) and let B be the subgroup of upper triangular matrices. Prove that G decomposes into exactly two double cosets of (B,B). (This is an example of the so-called *Bruhat decomposition* which is of great importance in the theory of algebraic groups and their representations.)<br />
<br />
c) Let G be the symmetric group S_n, and let H be the subgroup of permutations which fix 1. Describe the double coset decomposition of S_n into double cosets of (H,H).<br />
<br />
<br />
==HOMEWORK 3 (due Oct 2)==<br />
<br />
1. Compute the centralizer of the triple flip (12)(34)(56) in S_6. (Note: this is the kind of problem that a computer algebra package like sage is very good at, and I encourage you to learn to use sage well enough to check your answer.)<br />
<br />
2. Warning: a normal subgroup of a normal subgroup need not be normal! Let H be the subgroup of S_4 generated by (12)(34). Then H is a subgroup of the Klein 4-group V_4, which we have already shown is normal in S_4. Show that H is normal in V_4, but H is not normal in S_4.<br />
<br />
3. Let G be a group and let H_1 and H_2 be subgroups of G of finite index (that is, the number of cosets in G/H_i is finite, i = 1,2.) Prove that H_1 intersect H_2 also has finite index in G.<br />
<br />
4. Let X be the set of lines in F_3^2 (here a line means a 1-dimensional linear subspace) so that |X| = 4. Let G be the projective linear group PGL_2(F_3), so that G acts on X via its action on F_3^2. This action can be thought of as a homomorphism from G to S_4. Show that this homomorphism is an ISOMORPHISM from PGL_2(F_3) to S_4.<br />
<br />
5. We proved two theorems in class: Lagrange's theorem, which says that the order of a subgroup of G divides |G|, and Cauchy's theorem, which says that if p divides |G| then there exists a subgroup of G of order p. Cauchy's theorem is a kind of partial converse to Lagrange's theorem, but the full converse is false. Give an example of a finite group G and a divisor n of |G| such that you can prove there is no subgroup of G of order n.<br />
<br />
6. The group S_3 has only three conjugacy classes. Prove that a finite group with at most three conjugacy classes has order at most 6.<br />
<br />
7. Let X be the set of ordered triples of elements of {1,..,n}, for some n >= 3. Then S_n acts on X. How many orbits are there? (Hint: the answer does not depend on n.) OPTIONAL: How many orbits are there on the space of ordered k-tuples, when n >= k? <br />
<br />
<br />
<!--Below you will find a repository of homework problems. Note that some of these problems are taken from Lang's ''Algebra''.<br />
<br />
==HOMEWORK 1 (due Sep 20)==<br />
<br />
1. Suppose that H_1 and H_2 are subgroups of a group G. Prove that the intersection of H_1 and H_2 is a subgroup of G.<br />
<br />
2. Recall that S_3 (the symmetric group) is the group of permutations of the set {1..3}. List all the subgroups of S_3.<br />
<br />
3. We can define an equivalence relation on rational numbers by declaring two rational numbers to be equal whenever they differ by an integer. We denote the set of equivalence classes by Q/Z. The operation of addition makes Q/Z into a group.<br />
<br />
a) For each n, prove that Q/Z has a subgroup of order n.<br />
<br />
b) Prove that Q/Z is a ''divisible'' group: that is, if x is an element of Q/Z and n is an integer, there exists an element y of Q/Z such that ny = x. (Note that we write the operation in this group as addition rather than multiplication, which is why we write ny for the n-fold product of y with itself rather than y^n)<br />
<br />
c) Prove that Q/Z is not finitely generated. (Hint: prove that if x_1, .. x_d is a finite subset of Q/Z, the subgroup of Q/Z generated by x_1, ... x_d is finite.)<br />
<br />
4. Let F_2 be the free group on two generators x,y. Prove that there is a automorphism of F_2 which sends x to xyx and y to xy, and prove that this automorphism is unique.<br />
<br />
5. Let H be a subgroup of G, and let N_G(H), the ''normalizer'' of H in G, be the set of elements g in G satsifying g H g^{-1} = H. Prove that N_G(H) is a subgroup of G.<br />
<br />
6. Let T be the subgroup of GL_2(R) consisting of diagonal matrices. This is an example of a "Cartan subgroup," or "torus" (whence the notation T.) Describe the normalizer N(T) of T in GL_2(R). Prove that there is a homomorphism from N(T) to Z/2Z whose kernel is precisely T. <br />
<br />
7. Prove that the group of inner automorphisms of a group is normal in the full automorphism group Aut(G).<br />
<br />
8. Let H and H' be subgroups of a group G, and let x be an element of G. We denote by HxH' the set of elements of the form hxh', with h in H and h' in H', and we call HxH a ''double coset'' of the pair (H,H').<br />
<br />
a) Show that G decomposes as a disjoint union of double cosets of (H,H').<br />
<br />
b) Let G be GL_2(R) and let B be the subgroup of upper triangular matrices. Prove that G decomposes into exactly two double cosets of (B,B).<br />
<br />
c) Let G be the symmetric group S_n, and let H be the subgroup of permutations which fix 1. Describe the double coset decomposition of S_n into double cosets of (H,H).<br />
<br />
==HOMEWORK 2 (due Sep 27)==<br />
<br />
1. Prove that the symmetric group S_n is generated by transpositions. Hint: proceed by induction on n. Let H be the subgroup of S_n consisting of permutations fixing 1. Observe that H is isomorphic to S_{n-1}, so that H is generated by transpositions. Explain why it suffices to show that, for any g in S_n, the coset gH contains a transposition. Then explain why this is the case.<br />
<br />
2. Given that the symmetric group S_n is generated by transpositions, show that there are no nontrivial homomorphisms from S_n to Z/pZ for any odd prime p.<br />
<br />
3. Compute the centralizer of the triple flip (12)(34)(56) in S_6. (Note: this is the kind of problem that a computer algebra package like sage is very good at, and I encourage you to learn to use sage well enough to check your answer.)<br />
<br />
4. Warning: a normal subgroup of a normal subgroup need not be normal! Let H be the subgroup of S_4 generated by (12)(34). Then H is a subgroup of the Klein 4-group V_4, which we have already shown is normal in S_4. Show that H is normal in V_4, but H is not normal in S_4.<br />
<br />
5. The '''quaternion group''' Q is a non-abelian finite group of order 8. Its elements are 1, -1, i, -i, j, -j, k, -k; it satisfies g^2 = -1 for all g except +-1, and ijk = 1. (Note that the notation "-i" means "the product of i with -1" and -1 is understood to be central in Q.) Prove that every subgroup of Q is normal.<br />
<br />
6. (finishing example done in class) Let Gamma be the group F<x,y> / (x^2 = 1, y^2 = 1). (Remember, this means the quotient of the free group by the group generated by all conjugates of x^2 and y^2.) Prove that Gamma is infinite. Prove that Gamma has a subgroup R generated by xy isomorphic to Z, that R is normal, and that the quotient Gamma/R is isomorphic to Z/2Z. <br />
<br />
7. Suppose a group G has the property that every element in g satisfies g^2 = 1. Show that G must be abelian.<br />
<br />
==HOMEWORK 3 (due Oct 4)==<br />
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1. Multiplication by -1 is an automorphism of Z/nZ. Let a be the homomorphism Z/2Z -> Aut(Z/nZ) which sends the nontrivial element of Z/2Z to multiplication by -1. Then we have a semidirect product of Z/pZ by Z/2Z as defined in class. This group of order 2n is called a ''dihedral group.'' and is denoted D_n (or sometimes D_{2n}).<br />
<br />
1a. Compute the center of D_n. (Note that the answer depends on n!)<br />
1b. Show that all involutions of D_n are conjugate to each other if and only if n is odd.<br />
<br />
2. (More on direct products that we didn't do in class.) Suppose that G is a semidirect product of N with H. Suppose furthermore that H is also normal in G. Prove that G is a direct product of N with H.<br />
<br />
3. Let Q be the quaternion group from last week's homework. Show that Q does not decompose as a semidirect product N \rtimes H, apart form the trivial cases N = {e} and N = Q.<br />
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4. The ''affine linear group'' of degree n is the group of transformations from R^n to R^n of the form x -> Ax + b, where A is a matrix in GL_n(R) and b is a vector in R^n. Prove that the affine linear group is a semidirect product R^n \rtimes GL_n.<br />
<br />
5. The ''ordinary triangle group'' T(p,q,r) is the group with presentation <x,y | x^p = y^q = (xy)^r = 1>. This is a very interesting family of groups which arises naturally in many contexts in number theory and topology.<br />
<br />
6a. Prove that T(2,2,n) is isomorphic to the dihedral group D_n.<br />
6b. Prove that T(2,3,3) is finite and compute its order. Is it isomorphic to a finite group we've discussed in class?<br />
<br />
(It is a beautiful fact, but outside the scope of this course, that T(p,q,r) is finite if and only if 1/p + 1/q + 1/r > 1.)<br />
<br />
7. Let G be a subgroup of GL_n(R) which contains SL_n(R). Prove that G is a normal subgroup of GL_n(R).<br />
<br />
8. Let G be a group and let H_1 and H_2 be subgroups of G of finite index (that is, the number of cosets in G/H_i is finite, i = 1,2.) Prove that H_1 intersect H_2 also has finite index in G.<br />
<br />
9. Recall that the commutator [x,y] is x y x^{-1} y^{-1}. If G is a group, the commutator subgroup of G, denoted G' or [G,G], is the subgroup of G generated by all commutators [x,y] for x,y in G.<br />
<br />
9a. Show that G' is a normal subgroup of G.<br />
9b. Show that G/G' is an abelian group.<br />
9c. Show that if f: G -> A is a homomorphism, with A abelian, then G' is contained in ker f. Conclude that f factors through a homomorphism G/G' -> A.<br />
9d. Show that G' = G if and only if G has no nontrivial abelian quotient. In this case, we say G is ''perfect''.<br />
<br />
10. Under what conditions on (p,q,r) is the triangle group T(p,q,r) perfect?<br />
<br />
==HOMEWORK 4 (due Oct 16)==<br />
<br />
1. Let X be a set with a transitive action of a group G, which we think of as a homomorphism f: G -> Sym(X). Let H_x be the stabilizer of an element x of X.<br />
<br />
1a. If x' is another element of X, show that H_{x"} and H_x are conjugate subgroups. (Hint: this is not true without the hypothesis of transitivity, so make sure to use it!)<br />
<br />
1b. Show that the kernel of f is the intersection of all the conjugates of H_x in G. (This subgroup is called the ''normal core'' of H_x.)<br />
<br />
2. This week, a certified mathematical grown-up asked on MathOverflow what the index-3 subgroups of SL_2(Z/pZ) were. Let's do it ourselves for homework. Let p be a prime, and suppose H is an index-3 subgroup of SL_2(Z/pZ). I do not claim that the proof sketched below is necessarily the easiest, but it allows us to cover some ground that we won't get to do in lecture.<br />
<br />
2a. Using the coset action, show that the normal core of H is the kernel of a homomorphism f from SL_2(Z/pZ) to S_3.<br />
<br />
2b. A ''unipotent matrix'' is a matrix A such that (A-I)^n = 0 for some positive integer n. Prove that a unipotent element of SL_2(Z/pZ) has order dividing p.<br />
<br />
2c. You may use the fact that unipotent matrices generate SL_2(Z/pZ). Given this fact, show that f must be trivial when p > 3.<br />
<br />
2d. Using the above, show that SL_2(Z/pZ) has no index-3 subgroups when p > 3.<br />
<br />
2e. (extra credit) What can you say about the minimal index of a proper subgroup of SL_2(Z/pZ)?<br />
<br />
3. Let H be a subgroup of G of index 2. Prove that H is normal.<br />
<br />
4. We proved two theorems in class: Lagrange's theorem, which says that the order of a subgroup of G divides |G|, and Cauchy's theorem, which says that if p divides |G| then there exists a subgroup of G of order p. Cauchy's theorem is a kind of partial converse to Lagrange's theorem, but the full converse is false. Give an example of a finite group G and a divisor n of |G| such that you can prove there is no subgroup of G of order n.<br />
<br />
5. Let X be the set of lines in F_3^2 (here a line means a 1-dimensional linear subspace) so that |X| = 4. Let G be the projective linear group PGL_2(F_3), so that G acts on X via its action on F_3^2. This action can be thought of as a homomorphism from G to S_4. Show that this homomorphism is an ISOMORPHISM from PGL_2(F_3) to S_4.<br />
<br />
6. The group S_3 has only three conjugacy classes. Prove that a finite group with at most three conjugacy classes has order at most 6.<br />
<br />
7. Let X be the set of ordered triples of elements of {1,..,n}, for some n >= 3. Then S_n acts on X. How many orbits are there? (Hint: the answer does not depend on n.)<br />
<br />
8. Suppose that G and H are groups and f: G -> H is a homomorphism. Recall the definition of the abelianization G^ab from the previous problem set.<br />
<br />
8a. Show that the composition G -> H -> H^ab factors through a unique homomorphism G^ab -> H^ab, which we denote f^ab.<br />
<br />
8b. Show that if f: G -> H and g: H -> Q are homomorphisms, then f^ab o g^ab = (f o g)^ab.<br />
<br />
(For those reading MacLane, this constitutes a proof that abelianization is a '''functor''' from the category of groups to the category of abelian groups.)<br />
<br />
==HOMEWORK 5 (due Oct 23)==<br />
<br />
1. Show that there are only two different isomorphism classes of groups of order 14. (Hint: prove that such a group must be a semidirect product of a group of order 7 by a group of order 2.)<br />
<br />
2. (variant of Burnside's Lemma) Let G be a finite group and H a normal subgroup. Suppose G acts on a finite set X. <br />
<br />
2a. Show that if x and y lie in the same H-orbit, and g is an element of G, then gx and gy also lie in the same H-orbit. Thus, G/H has a natural action on the set of H-orbits X/H.<br />
<br />
2b. Show that the average number of fixed points in X of an element of the coset gH is the same as the number of fixed points of the element gH of G/H in its action on X/H.<br />
<br />
3. A ''central extension'' of a group G by an abelian group A is a group E, together with a surjective homomorphism E -> G whose kernel is central in E and is isomorphic to A.<br />
<br />
3a. Show that the quaternion group is a central extension of (Z/2Z)^2 by Z/2Z.<br />
<br />
3b. Give an example of another nonabelian group which is a central extension of (Z/2Z)^2 by Z/2Z and which is not isomorphic to the quaternion group.<br />
<br />
4. Prove that a central extension of an abelian group is nilpotent.<br />
<br />
5. Give two different composition series for S_4 and show that they have the same composition factors.<br />
<br />
6. Suppose that A is a finitely generated abelian group, and B is a subgroup of A.<br />
<br />
6a. Prove that B is finitely generated and that rank(B) <= rank(A).<br />
<br />
6b. Prove that rank(A/B) = rank(A) - rank(B).<br />
<br />
7. For this problem, you may use the fact (which I should have asked on homework earlier) that if H_1 and H_2 are subgroups of G, then<br />
<br />
[G: H_1 intersect H_2] <= [G:H_1][G:H_2].<br />
<br />
Prove that if G is a simple group, d is an integer greater than 1, and S is a subgroup of index d in G, then |G| <= d^d.<br />
<br />
Conclude that, in particular, A_5 has no subgroup of index 3. On the other hand, exhibit a subgroup of A_5 of index 5.<br />
<br />
(Note: d^d is not a sharp bound; can you get a bound of d factorial instead?)<br />
<br />
<!--(This argument shows that a solvable subgroup of A_n has size at most about (n-1)!, but this is far from sharp; e.g. I found a citation to a 1967 paper of Dixon which shows that a solvable subgroup of A_n has size at most k^n where k is around 2.88.)--><br />
<br />
<!--<br />
8. Give an example of an abelian group A with a subgroup B such that B is not a direct summand of A.<br />
<br />
9. Let A be the subgroup of Z^2 generated by the vector (1,0). Let B be the subgroup of Z^2 generated by the vector (3,3). Show that A and B are isomorphic to each other, but Z^2 / A is not isomorphic to Z^2 / B.<br />
<br />
==HOMEWORK 6 (due Oct 30)==<br />
<br />
1. In this problem, you will construct a functor F from the category of finite sets to the category of complex vector spaces. I'll tell you what it does on objects: if X is a finite set, then F(X) is the vector space spanned by a set of basis elements {e_x}_{x in X}, so that dim F(X) = |X|. Your job: complete the definition of F by describing the map from Mor_{Finite Sets}(X,Y) to Mor_{Vector Spaces}(F(X),F(Y)), and showing that your map respects composition.<br />
<br />
2. If (V, rho) is a representation of a group G, denote by V^G the space of vectors in V such that gv = v for all g in G; these are called _invariant_ vectors. We saw that the permutation representation of S_3 has a one-dimensional space of invariants. Prove that if X is a finite set with G-action, and V_X the corresponding permutation representation of G, then<br />
<br />
dim V_X^G = number of orbits of X.<br />
<br />
3. (Invariants are functorial) Let G be a group. Then there is a category called C[G]-Mod whose objects are complex representations of G (i.e. pairs (V,rho) where V is a complex vector space and rho is a homomorphism from G to GL(V)). The morphisms in this category are the ones described in class: Mor((V,rho),(W,psi)) is the set of linear maps f from V to W such that<br />
<br />
f(rho(g)(v)) = psi(g)(f(v))<br />
<br />
for all g in G and all v in V.<br />
<br />
Show that there is a functor H_0 from the category of representations of G to the category of vector spaces whose action on objects is given by<br />
<br />
H_0((V) = V^G.<br />
<br />
(In other words: your job again is to explain how to associate to a morphism from (V,rho) to (W,phi) a morphism from V^G to W^G, in a way that's compatible with composition.)<br />
<br />
<!--4. The group PSL_2(Z) has a presentation of the form [U,V|U^2 = V^3 = 1]. Thus, an n-dimensional representation of PSL_2(Z) is precisely a pair of n x n matrices (rho(U),rho(V)), such that rho(U)^2 = rho(V)^3 = 1. Prove that for every n > 1 there are uncountably many isomorphism classes of n-dimensional representations of PSL_2(Z). (Extra credit: show that there are uncountably many isomorphism classes of _irreducible_ representations.)<br />
<br />
An amazing theorem of Margulis (superrigidity) shows that SL_2(Z) is very special among arithmetic groups in this way. For instance, there are only _finitely_ many isomorphism classes of n-dimensional representations of PSL_d(Z) for any d > 2; those groups are "rigid."--><br />
<br />
<!--<br />
5. Let G be a finite group, let X be a set with G-action, and let chi be the character of the corresponding permutation representation. Show that, for each g in G, chi(g) is the number of elements of X fixed by g. (I am actually going to prove this in class.)<br />
<br />
6. Let V be the space of homogeneous degree-3 polynomials in variables x_1, x_2, x_3; then S_3 acts on V by permutation of the three variables. Describe the decomposition of V into irreducible representations of S_3.<br />
<br />
7a. Show that if chi is the character of a representation of a finite group G, then chi(g^{-1}) is the complex conjugate of chi.<br />
<br />
7b. As a consequence, show that if g is conjugate to g^{-1} for all g in G, then chi(g) is real.<br />
<br />
7c. The groups D_p and S_n have the property that g is conjugate to g^{-1} for _every_ g in G, so that every character is a real-valued function on G. Give an example of a group G where this is not the case, and give an example of a representation of G whose character takes non-real values.<br />
<br />
==HOMEWORK 7 (due Nov 6)==<br />
<br />
1. Let G be a finite group, let X be a finite set with G-action, and let chi be the character of the corresponding permutation representation. Show that, for each g in G, chi(g) is the number of elements of X fixed by g. (I am actually going to prove this in class.)<br />
<br />
2a. Show that if chi is the character of a finite-dimensional representation of a finite group G, then chi(g^{-1}) is the complex conjugate of chi(g) for all g.<br />
<br />
2b. As a consequence, show that if g is conjugate to g^{-1} for all g in G, then chi(g) is real.<br />
<br />
2c. The groups D_p and S_n have the property that g is conjugate to g^{-1} for _every_ g in G, so that every character is a real-valued function on G. Give an example of a group G where this is not the case, and give an example of a representation of G whose character takes non-real values.<br />
<br />
3. Let V and W be complex vector spaces. Let f be a linear transformation of V, and g a linear transformation of W. <br />
<br />
3a. Show that there is a unique linear transformation F satisfying<br />
<br />
F(v tensor w) = f(v) tensor g(w)<br />
<br />
for all v in V and all w in W. We denote this transformation by f tensor g.<br />
<br />
3b. Suppose V and W are finite dimensional. Show that Trace(f tensor g) = Trace(f) Trace(g). What does this say when f and g are both the identity transformation?<br />
<br />
4. Let V be a vector space. We define Sym^2 V to be the quotient of V tensor V by the subspace generated by all elements of the form<br />
<br />
v tensor w - w tensor v<br />
<br />
for v,w in V.<br />
<br />
Suppose dim V = n. What is dim Sym^2 V?<br />
<br />
5. Show that Z/pZ tensor Z/qZ is zero when p and q are distinct primes.<br />
<br />
==HOMEWORK 8 (due Nov 13)==<br />
<br />
1. (Schur's lemma can be false over non-algebraically closed fields) Let G be Z/3Z and let g be a generator of G. Then G acts by cyclic permutations on the set {1,2,3}; let P be the corresponding permutation representation on the REAL (not complex!) vector space generated by e_1,e_2,e_3.<br />
<br />
1a. Show that P breaks up as the direct sum of the trivial representation and a 2-dimensional representation V which is irreducible. (Remember, we are not working over the complex numbers, so you cannot use the criterion for irreducibility in terms of the character that we worked out in class for the complex case; you need to prove directly that V has no nontrivial proper subrepresentation.)<br />
<br />
1b. Show that the map from V to V induced by g is G-equivariant, but is neither zero nor a scalar multiplication.<br />
<br />
2. If V_1 and V_2 are representations (not necessarily irreducible!) of G, show that <chi_V_1, chi V_2> is a non-negative integer.<br />
<br />
3. When a formula always yields a non-negative integer, it is often because it is secretly a formula for either the cardinality of some finite set or the dimension of some finite-dimensional vector space. In this case, show that<br />
<br />
<chi_{V_1}, chi_{V_2}> = dim_C Hom_G(V_1,V_2)<br />
<br />
where we recall that Hom_G(V_1,G_2) refers to the space of G-equivariant homomorphisms from V_1 to V_2.<br />
<br />
4. If A is an abelian group, show that every irreducible representation of A is 1-dimensional. (Hint: show that there is a vector which is an eigenvector for every element of A!)<br />
<br />
5. Let G be the group of order 20 which is the semidirect product of A by H, where A is Z/5Z and H is (Z/5Z)^* with its natural action on A. We will work out all the irreducible representations of A. <br />
<br />
5a. First of all, consider the 5-dimensional permutation representation of G acting by left multiplication on the cosets of H. Show that this representation decomposes as a direct sum of the trivial representation and an irreducible 4-dimensional representation. (Hint: it is probably easier to use the character of this representation than to prove irreducibility directly.) <br />
<br />
5b. Now show that there are 4 1-dimensional representations of G which factor through the quotient G/A (i.e. such that rho(A) is the identity.)<br />
<br />
5c. Using the decomposition of the regular representation, show that the irreducible representations of G you have constructed are the only ones.<br />
<br />
==HOMEWORK 9 (due Nov 20)==<br />
<br />
0. Find all ring homomorphisms from Z to Z/30Z. [Warning: there are 8 of them, not 30.]<br />
<br />
<br />
1. If R is a ring (not necessarily with 1) such that a^2 = a for every a in R, R is called a Boolean ring.<br />
<br />
1a. Show that any nonzero Boolean ring has characteristic 2. (In other words, show that a+a=0 for all a in R.)<br />
<br />
1b. Show that every Boolean ring is commutative.<br />
<br />
<br />
2. Let phi : R -> S be a homomorphism of commutative rings with 1.<br />
<br />
2a. If I is an ideal of S, show that phi^(-1) (I) is an ideal of R.<br />
<br />
2b. If P is a prime ideal of S, show that phi^(-1) (P) is either R or a prime ideal of R.<br />
<br />
2c. If M is a maximal ideal of S and phi is surjective, show that phi^(-1) (M) is a maximal ideal of R.<br />
<br />
2d. Find a homomorphism phi : R -> S and a maximal ideal M of S such that phi^(-1) (M) is not a maximal ideal of R.<br />
<br />
<br />
3. Let Nil(A) denote the set of nilpotent elements of a ring A, and let R be a commutative ring. In class it was observed that Nil(R) is an ideal of R.<br />
<br />
3a. Find (a generator of) Nil(Z/720Z).<br />
<br />
3b. Show that Nil(R/Nil(R)) = 0.<br />
<br />
3c. If x is in Nil(R) and u is a unit, show that u+x is a unit.<br />
<br />
3d. Show that Nil(S), where S is the 2x2 matrices with real coefficients, is not an ideal of S.<br />
<br />
<br />
4a. If M is an R-module and I is an ideal of R such that im = 0 for all i in I and m in M, show that M carries the structure of an R/I-module.<br />
<br />
4b. If S is a commutative ring with 1 and J is a maximal ideal of S, show that J/J^2 is a vector space over S/J.<br />
<br />
<br />
5. A "divisible element" x of a Z-module is one such that for each positive integer k, there exists a y such that ky=x. Show that a free Z-module cannot contain a nonzero divisible element.<br />
<br />
<br />
6. Prove that the direct sum of any collection of free R-modules is also a free R-module.<br />
<br />
<br />
7. In this problem you will prove that the module M given by the direct product of copies of Z, indexed by the positive integers, is not a free Z-module. Therefore, assume that M is a free Z-module with basis B, and let N be the direct sum of copies of Z, as a submodule of M.<br />
<br />
7a. Show there is a countable subset B1 of B such that N is contained in the submodule N1 generated by B1. [Hint: N is countable.]<br />
<br />
7b. Show that M/N1 is a free Z-module.<br />
<br />
7c. Let S = {(b1, b2, b3, ...) : b_i = i! or -i!}. Show that there is some s in S which is not in N1. [Hint: S is uncountable.]<br />
<br />
7d. For any s in S but not N1, show that for each positive integer k, there exists some m in M with s-km in N1. Conclude that s+N1 is a divisible element of M/N1.<br />
<br />
7e. Use problem 5 to obtain a contradiction.<br />
<br />
<br />
8. In class I said "rank" is not always well-defined for R-modules if R is not commutative. Here is an example demonstrating this assertion. Let M be the countably infinite direct product of copies of Z as in exercise 7, and let R be the endomorphism ring of M. Define phi_1(a1,a2,a3,a4,...) = (a1,a3,a5,...) and phi_2(a1,a2,a3,a4,...) = (a2,a4,a6,...).<br />
<br />
8a. Show that {phi_1, phi_2} is a free basis for R, as a left R-module. [Hint: let psi_1(a1,a2,a3,...) = (a1,0,a2,0,...) and psi_2(a1,a2,a3,...) = (0,a1,0,a2,...). Compute the products of the phi_i and psi_j in either order, and use the result to show that the phi_i are a basis.]<br />
<br />
8b. Show that R is isomorphic to R^n for every positive integer n.<br />
<br />
<br />
==HOMEWORK 10 (due Dec 4)==<br />
<br />
1. '''How algebraists do calculus'''. The ''ring of dual numbers'' is the ring R = C[e]/(e^2). We think of e as an infinitesimal number; that is, something so small that its square is 0. This algebraic notion of infinitesimal is the basis for the algebraic notion of differentiation, which works as follows. Let f be a polynomial in R[x]. Show that there is a unique polynomial g in C[x] such that<br />
<br />
f(x+e) - f(x) = eg(x)<br />
<br />
and that in fact g is the derivative of f. In other words, the usual definition of "derivative" works just fine in this context, without any use of the notion of limit!<br />
<br />
2. Write down all the ideals of C[e]/e^2. (There are three.)<br />
<br />
3. Show that any right ideal of the ring M_2(Q) of 2x2 matrices is an ideal of the form I_V described in class for some subspace V of Q^2; that is, it consists precisely of those linear transformations whose image is contained in V.<br />
<br />
<br />
4. An ''idempotent'' element of a ring is an element e satisfying e^2 = e.<br />
<br />
4a. If e is an idempotent, show that 1-e is also an idempotent.<br />
<br />
4b. Suppose R has a unit (we are actually always supposing this, but I just want to remind you of it for this problem) and suppose e is a ''central'' idempotent, i.e. an idempotent which commutes with all elements of R. Show that eR and (1-e)R are both two-sided ideals. Moreover, show that eR and (1-e)R are also subrings of R (with the warning that the identity in eR is not 1, which is not contained in eR, but rather e.)<br />
<br />
4c. Show that the map f_e: R -> eR defined by f_e(r) = er is a ring homomorphism. Finally show that the ring homomorphism<br />
<br />
f_{e} x f_{1-e}: R -> eR x (1-e)R <br />
<br />
is actually an isomorphism of rings. In other words, the presence of a central idempotent induces a decomposition a ring as a direct product.<br />
<br />
<br />
5. The ''center'' of a ring R is the subring of elements commuting with every element of R.<br />
<br />
5a. Show that the center of M_n(Q) is the ring of scalar matrices.<br />
<br />
5b. Show that the center of the group ring C[G] is the set of elements sum f(g) g such that f(g) is a class function.<br />
<br />
<br />
6. '''How algebraists do differential equations.''' We define the ''Weyl algebra'' W of differential operators in one variable as follows. Multiplication by x is a linear transformation on the complex vector space V of all smooth functions on C. Differentiation, which we denote d, is also a linear transformation on V. So we let W be the subring of End(V) generated by x and d. We note that x and d satisfy the relation xd = dx - 1; in particular, the Weyl algebra is not commutative.<br />
<br />
6a. Describe the submodule of V which is annihilated by the element d-x in W.<br />
<br />
6b. Show that, for any polynomial f(x), fd-df = -f', and also show that for any polynomial g(d), gx-xg = -g'. (This is another way algebraists do calculus!) <br />
<br />
6c. Use the facts from (b) to show that the center of the Weyl algebra is just the field of constants. (Remark: you will use in the course of this proof the fact that a polynomial whose derivative is 0 is constant -- this would not be true if we were working in positive characteristic, as the example of x^p considered as a polynomial over F_p demonstrates.)<br />
<br />
<br />
7. If M is a left module for a ring R, we denote by Ann(M) the set of elements r in R such that rm = 0 for all m in M. Show that Ann(M) is a two-sided ideal. <br />
<br />
<br />
8. Let M be a left module for R, and let m be an element of M. We denote by Ann(m) the set of elements r in R such that rm = 0.<br />
<br />
8a. Show that Ann(m) is a left ideal, and show that the submodule Rm of M generated by m is isomorphic to the left module R/Ann(m).<br />
<br />
8c. Let V,W be as in question 6. Show that Ann(e^x) is the left ideal W(d-1).<br />
<br />
==HOMEWORK 11 (due Dec 13)==<br />
<br />
1. Let f: R -> S be a homomorphism of rings. As discussed in class, if M is a left R-module, then S tensor_R M is a left S-module, and if N is an S-module, then N can also be considered as an R-module by pullback of the action.<br />
<br />
Prove that if M is a finitely generated left R-module, then S tensor_R M is a finitely generated left S-module. On the other hand, give an example to show that N can be finitely generated as a left S-module but not finitely generated when considered as a left R-module.<br />
<br />
2. Recall from class that when V and W are complex vector spaces, W^dual tensor V is naturally isomorphic to the space of linear homomorphisms Hom(W,V).<br />
<br />
Now suppose that V and W are left C[G]-modules, i.e. complex representations of a group G. Show that W^dual tensor_C[G} V is the maximal G-invariant quotient of Hom(W,V); in other words, the quotient of Hom(W,V) by the subspace generated by all functions F of the form F(w) = f(gw) - gf(w) for some function f in Hom(W,V) and some g in G.<br />
<br />
3. '''How algebraists do logic.''' Why are Boolean rings called Boolean rings? Because they model some features of the Boolean arithmetic that governs propositional logic. Let A be a collection of propositions. We define two operations on A as follows. By a + b we mean the negation of a xor b, while by ab we mean "a or b."<br />
<br />
3a. It is not a priori obvious that these operations satisfy the ring axioms, but assume this for the moment. Show that a^2 = a for all a (so A is a boolean ring) and 2a = 0 for all a (so A has characteristic 2.)<br />
<br />
3b. Suppose we have a consistent assignment of truth-values to the propositions in A. Show that the set of true statements of A forms an ideal of A.<br />
<br />
3c. (Optional for people who like propositional logic) Show that a(b+c) = ab + ac with the given operations.<br />
<br />
<br />
4. Let K be Q[x]/(x^3-2).<br />
<br />
4a. Show that K is a field.<br />
<br />
4b. Show that K tensor_Q K decomposes as the direct sum of two fields. What are they? [Note: K tensor_Q K is a ring!]<br />
<br />
5. There was some discussion in class about when we can "go back and forth" between right modules and left modules, as is the case with group rings.<br />
<br />
5a. If R is a ring, we denote R^op ("the opposite ring") to be the ring whose elements are those of R, but whose multiplication law x_opp is defined by s x_opp r = rs. Show that if M is a left R-module, then the map M x R^op -> M sending (x,r) to rx makes M into a right R^op module. In particular, if there is a RING ISOMORPHISM from R to R^op, then every left R-module can be thought of as a right R-module.<br />
<br />
5b. Verify that the map sending g to g^{-1} extends by linearity to give a ring isomorphism between C[G] and C[G]^op.<br />
<br />
5c. Construct a ring isomorphism between M_n(Q) and M_n(Q)^op.<br />
<br />
5d. Optional: Give an example of a ring (with 1) which is not isomorphic to its opposite ring. (This is not so easy, I think! I found a MathOverflow thread with several examples, none of which came with a short proof of the non-isomorphism -- try your hand!)<br />
<br />
6. . Let R be a commutative ring with 1. If R has a unique maximal ideal M, R is called a "local ring".<br />
<br />
6a. If R is a local ring, show that every element of R that is not in M is a unit.<br />
<br />
6b. Conversely, if R is such that the set of nonunits forms an ideal I, show that R is a local ring with maximal ideal I.<br />
<br />
7. The inclusion of S_3 into S_4 associated to the inclusion of {1,2,3} into {1,2,3,4} induces a ring homomorphism from C[S_3} to C{S_4]. Let V be a representation of C{S_3]. Then, as discussed in class, V tensor_C[S_3] C[S_4] is a representation of S_4, called the '''induced representation''' of V.<br />
<br />
7a. Show that dim_C (V tensor_C[S_3] C[S_4]) = 4 dim_C V.<br />
<br />
7b. When V is the trivial representation of S_3, decompose (V tensor_C[S_3] C[S_4]) as a sum of irreducible representations of S_4.<br />
<br />
--><br />
<br />
<!--==HOMEWORK PROBLEMS TO GIVE IN FUTURE SEMESTERS==<br />
<br />
Let f: C[x] tensor_C C[y] -> C[z] be the map of rings with f(x) = f(y) = z. Show that f(m tensor n) is not 0 for any pure tensor m tensor n, but f is not injective. (This is a warning not to try to check injectivity by checking that no pure tensors are killed by the map.)<br />
<br />
--></div>Dummithttps://hilbert.math.wisc.edu/wiki/index.php?title=741&diff=58617412013-09-11T19:59:56Z<p>Dummit: </p>
<hr />
<div>'''Math 741'''<br />
<br />
Algebra<br />
<br />
Prof: [http://www.math.wisc.edu/~ellenber Jordan Ellenberg]<br />
<br />
Grader: Evan Dummit. Late homework may be given directly to the grader, along with either (i) the instructor's permission, or (ii) a polite request for mercy.<br />
<br />
Ellenberg's office hours: Friday 3pm.<br />
<br />
This course, the first semester of the introductory graduate sequence in algebra, will cover the basic theory of groups, group actions, representation, linear and multilinear algebra, and the beginnings of ring theory.<br />
<br />
==SYLLABUS==<br />
<br />
In this space we will record the theorems and definitions we covered each week, which we can use as a list of notions you should be prepared to answer questions about on the Algebra qualifying exam. The material covered on the homework is also an excellent guide to the scope of the course. <br />
<br />
'''WEEK 1''': <br />
<br />
Definition of group. Associativity. Inverse. <br />
<br />
Examples of group: GL_n(R). GL_n(Z). Z/nZ. R. Z. R^*. The free group F_k on k generators. <br />
<br />
Homomorphisms. The homomorphisms from F_k to G are in bijection with G^k. Isomorphisms.<br />
<br />
'''WEEK 2''':<br />
<br />
The symmetric group (or permutation group) S_n on n letters. Cycle decomposition of a permutation. Order of a permutation. Thm: every element of a finite group has finite order. <br />
<br />
Subgroups. Left and right cosets. Lagrange's Theorem. Cyclic groups. The order of an element of a finite group is a divisor of the order of the group.<br />
<br />
The sign homomorphism S_n -> +-1.<br />
<br />
<br />
'''WEEK 3'''<br />
<br />
Normal subgroups. The quotient of a group by a normal subgroup. The first isomorphism theorem. Examples of S_n -> +-1 and S_4 -> S_3 with kernel V_4, the Klein 4-group.<br />
<br />
Centralizers and centers. Abelian groups. The center of SL_n(R) is either 1 or +-1.<br />
<br />
Groups with presentations. The infinite dihedral group <x,y | x^2 = 1, y^2 = 1>.<br />
<br />
'''WEEK 4'''<br />
<br />
More on groups with presentations.<br />
<br />
Second and third isomorphism theorems.<br />
<br />
Semidirect products.<br />
<br />
'''WEEK 5'''<br />
<br />
Group actions, orbits, and stabilizers.<br />
<br />
Orbit-stabilizer theorem.<br />
<br />
Cayley's theorem.<br />
<br />
Cauchy's theorem.<br />
<br />
'''WEEK 6'''<br />
<br />
Applications of orbit-stabilizer theorem (p-groups have nontrivial center, first Sylow theorem.)<br />
<br />
Classification of finite abelian groups and finitely generated abelian groups.<br />
<br />
Composition series and the Jordan-Holder theorem (which we state but don't prove.) <br />
<br />
The difference between knowing the composition factors and knowing the group (e.g. all p-groups of the same order have the same composition factors.)<br />
<br />
'''WEEK 7'''<br />
<br />
Simplicity of A_n.<br />
<br />
Nilpotent groups (main example: the Heisenberg group)<br />
<br />
Derived series and lower central series.<br />
<br />
Category theory 101: Definition of category and functor. Some examples. A group is a groupoid with one object.<br />
<br />
'''WEEK 8'''<br />
<br />
Introduction to representation theory.<br />
<br />
'''WEEK 10'''<br />
<br />
Ring theory 101: Rings, ring homomorphisms, ideals, isomorphism theorems. Examples: fields, Z, the Hamilton quaternions, matrix rings, rings of polynomials and formal power series, quadratic integer rings, group rings. Integral domains. Maximal and prime ideals. The nilradical.<br />
<br />
Module theory 101: Modules, module homomorphisms, submodules, isomorphism theorems. Noetherian modules and Zorn's lemma. Direct sums and direct products of arbitrary collections of modules.<br />
<br />
<br />
==HOMEWORK 1 (due Sep 16)==<br />
<br />
1. Suppose that H_1 and H_2 are subgroups of a group G. Prove that the intersection of H_1 and H_2 is a subgroup of G.<br />
<br />
2. Recall that S_3 (the symmetric group) is the group of permutations of the set {1..3}. List all the subgroups of S_3.<br />
<br />
3. We can define an equivalence relation on rational numbers by declaring two rational numbers to be equal whenever they differ by an integer. We denote the set of equivalence classes by Q/Z. The operation of addition makes Q/Z into a group.<br />
<br />
a) For each n, prove that Q/Z has a subgroup of order n.<br />
<br />
b) Prove that Q/Z is a ''divisible'' group: that is, if x is an element of Q/Z and n is an integer, there exists an element y of Q/Z such that ny = x. (Note that we write the operation in this group as addition rather than multiplication, which is why we write ny for the n-fold product of y with itself rather than y^n)<br />
<br />
c) Prove that Q/Z is not finitely generated. (Hint: prove that if x_1, .. x_d is a finite subset of Q/Z, the subgroup of Q/Z generated by x_1, ... x_d is finite.)<br />
<br />
d) Conclude that Q is not finitely generated.<br />
<br />
4. We will prove that there is no homomorphism from SL_2(Z) to Z except the one which sends all of SL_2(Z) to 0. Suppose f is a homomorphism from SL_2(Z) to Z.<br />
<br />
a) Let U1 be the upper triangular matrix with 1's on the diagonal and a 1 in the upper right hand corner, as in class, and let U2 be the transpose of U1, also as in class. Show that (U1 U2^{-1})^6 = identity (JING 1, TAO 0) and explain why this implies that f(U1) = f(U2).<br />
<br />
b) Show that there is a matrix A in SL_2(Z) such that A U1 A^{-1} = U2^{-1}. (Recall that we say U1 and U2^{-1} are "conjugate".) Explain why this also implies that f(U1) = -f(U2).<br />
<br />
c) Explain why a) and b) imply that f must be identically 0.<br />
<br />
5. The argument above also shows that there is no nonzero homomorphism from SL_2(Z) to Z/pZ where p is a prime greater than 2. However, it leaves open the possibility that there is indeed a nonzero homomorphism from SL_2(Z) to Z/2Z. Exhibit such a homomorphism.<br />
<br />
<br />
<br />
<br />
<br />
<!--Below you will find a repository of homework problems. Note that some of these problems are taken from Lang's ''Algebra''.<br />
<br />
==HOMEWORK 1 (due Sep 20)==<br />
<br />
1. Suppose that H_1 and H_2 are subgroups of a group G. Prove that the intersection of H_1 and H_2 is a subgroup of G.<br />
<br />
2. Recall that S_3 (the symmetric group) is the group of permutations of the set {1..3}. List all the subgroups of S_3.<br />
<br />
3. We can define an equivalence relation on rational numbers by declaring two rational numbers to be equal whenever they differ by an integer. We denote the set of equivalence classes by Q/Z. The operation of addition makes Q/Z into a group.<br />
<br />
a) For each n, prove that Q/Z has a subgroup of order n.<br />
<br />
b) Prove that Q/Z is a ''divisible'' group: that is, if x is an element of Q/Z and n is an integer, there exists an element y of Q/Z such that ny = x. (Note that we write the operation in this group as addition rather than multiplication, which is why we write ny for the n-fold product of y with itself rather than y^n)<br />
<br />
c) Prove that Q/Z is not finitely generated. (Hint: prove that if x_1, .. x_d is a finite subset of Q/Z, the subgroup of Q/Z generated by x_1, ... x_d is finite.)<br />
<br />
4. Let F_2 be the free group on two generators x,y. Prove that there is a automorphism of F_2 which sends x to xyx and y to xy, and prove that this automorphism is unique.<br />
<br />
5. Let H be a subgroup of G, and let N_G(H), the ''normalizer'' of H in G, be the set of elements g in G satsifying g H g^{-1} = H. Prove that N_G(H) is a subgroup of G.<br />
<br />
6. Let T be the subgroup of GL_2(R) consisting of diagonal matrices. This is an example of a "Cartan subgroup," or "torus" (whence the notation T.) Describe the normalizer N(T) of T in GL_2(R). Prove that there is a homomorphism from N(T) to Z/2Z whose kernel is precisely T. <br />
<br />
7. Prove that the group of inner automorphisms of a group is normal in the full automorphism group Aut(G).<br />
<br />
8. Let H and H' be subgroups of a group G, and let x be an element of G. We denote by HxH' the set of elements of the form hxh', with h in H and h' in H', and we call HxH a ''double coset'' of the pair (H,H').<br />
<br />
a) Show that G decomposes as a disjoint union of double cosets of (H,H').<br />
<br />
b) Let G be GL_2(R) and let B be the subgroup of upper triangular matrices. Prove that G decomposes into exactly two double cosets of (B,B).<br />
<br />
c) Let G be the symmetric group S_n, and let H be the subgroup of permutations which fix 1. Describe the double coset decomposition of S_n into double cosets of (H,H).<br />
<br />
==HOMEWORK 2 (due Sep 27)==<br />
<br />
1. Prove that the symmetric group S_n is generated by transpositions. Hint: proceed by induction on n. Let H be the subgroup of S_n consisting of permutations fixing 1. Observe that H is isomorphic to S_{n-1}, so that H is generated by transpositions. Explain why it suffices to show that, for any g in S_n, the coset gH contains a transposition. Then explain why this is the case.<br />
<br />
2. Given that the symmetric group S_n is generated by transpositions, show that there are no nontrivial homomorphisms from S_n to Z/pZ for any odd prime p.<br />
<br />
3. Compute the centralizer of the triple flip (12)(34)(56) in S_6. (Note: this is the kind of problem that a computer algebra package like sage is very good at, and I encourage you to learn to use sage well enough to check your answer.)<br />
<br />
4. Warning: a normal subgroup of a normal subgroup need not be normal! Let H be the subgroup of S_4 generated by (12)(34). Then H is a subgroup of the Klein 4-group V_4, which we have already shown is normal in S_4. Show that H is normal in V_4, but H is not normal in S_4.<br />
<br />
5. The '''quaternion group''' Q is a non-abelian finite group of order 8. Its elements are 1, -1, i, -i, j, -j, k, -k; it satisfies g^2 = -1 for all g except +-1, and ijk = 1. (Note that the notation "-i" means "the product of i with -1" and -1 is understood to be central in Q.) Prove that every subgroup of Q is normal.<br />
<br />
6. (finishing example done in class) Let Gamma be the group F<x,y> / (x^2 = 1, y^2 = 1). (Remember, this means the quotient of the free group by the group generated by all conjugates of x^2 and y^2.) Prove that Gamma is infinite. Prove that Gamma has a subgroup R generated by xy isomorphic to Z, that R is normal, and that the quotient Gamma/R is isomorphic to Z/2Z. <br />
<br />
7. Suppose a group G has the property that every element in g satisfies g^2 = 1. Show that G must be abelian.<br />
<br />
==HOMEWORK 3 (due Oct 4)==<br />
<br />
1. Multiplication by -1 is an automorphism of Z/nZ. Let a be the homomorphism Z/2Z -> Aut(Z/nZ) which sends the nontrivial element of Z/2Z to multiplication by -1. Then we have a semidirect product of Z/pZ by Z/2Z as defined in class. This group of order 2n is called a ''dihedral group.'' and is denoted D_n (or sometimes D_{2n}).<br />
<br />
1a. Compute the center of D_n. (Note that the answer depends on n!)<br />
1b. Show that all involutions of D_n are conjugate to each other if and only if n is odd.<br />
<br />
2. (More on direct products that we didn't do in class.) Suppose that G is a semidirect product of N with H. Suppose furthermore that H is also normal in G. Prove that G is a direct product of N with H.<br />
<br />
3. Let Q be the quaternion group from last week's homework. Show that Q does not decompose as a semidirect product N \rtimes H, apart form the trivial cases N = {e} and N = Q.<br />
<br />
4. The ''affine linear group'' of degree n is the group of transformations from R^n to R^n of the form x -> Ax + b, where A is a matrix in GL_n(R) and b is a vector in R^n. Prove that the affine linear group is a semidirect product R^n \rtimes GL_n.<br />
<br />
5. The ''ordinary triangle group'' T(p,q,r) is the group with presentation <x,y | x^p = y^q = (xy)^r = 1>. This is a very interesting family of groups which arises naturally in many contexts in number theory and topology.<br />
<br />
6a. Prove that T(2,2,n) is isomorphic to the dihedral group D_n.<br />
6b. Prove that T(2,3,3) is finite and compute its order. Is it isomorphic to a finite group we've discussed in class?<br />
<br />
(It is a beautiful fact, but outside the scope of this course, that T(p,q,r) is finite if and only if 1/p + 1/q + 1/r > 1.)<br />
<br />
7. Let G be a subgroup of GL_n(R) which contains SL_n(R). Prove that G is a normal subgroup of GL_n(R).<br />
<br />
8. Let G be a group and let H_1 and H_2 be subgroups of G of finite index (that is, the number of cosets in G/H_i is finite, i = 1,2.) Prove that H_1 intersect H_2 also has finite index in G.<br />
<br />
9. Recall that the commutator [x,y] is x y x^{-1} y^{-1}. If G is a group, the commutator subgroup of G, denoted G' or [G,G], is the subgroup of G generated by all commutators [x,y] for x,y in G.<br />
<br />
9a. Show that G' is a normal subgroup of G.<br />
9b. Show that G/G' is an abelian group.<br />
9c. Show that if f: G -> A is a homomorphism, with A abelian, then G' is contained in ker f. Conclude that f factors through a homomorphism G/G' -> A.<br />
9d. Show that G' = G if and only if G has no nontrivial abelian quotient. In this case, we say G is ''perfect''.<br />
<br />
10. Under what conditions on (p,q,r) is the triangle group T(p,q,r) perfect?<br />
<br />
==HOMEWORK 4 (due Oct 16)==<br />
<br />
1. Let X be a set with a transitive action of a group G, which we think of as a homomorphism f: G -> Sym(X). Let H_x be the stabilizer of an element x of X.<br />
<br />
1a. If x' is another element of X, show that H_{x"} and H_x are conjugate subgroups. (Hint: this is not true without the hypothesis of transitivity, so make sure to use it!)<br />
<br />
1b. Show that the kernel of f is the intersection of all the conjugates of H_x in G. (This subgroup is called the ''normal core'' of H_x.)<br />
<br />
2. This week, a certified mathematical grown-up asked on MathOverflow what the index-3 subgroups of SL_2(Z/pZ) were. Let's do it ourselves for homework. Let p be a prime, and suppose H is an index-3 subgroup of SL_2(Z/pZ). I do not claim that the proof sketched below is necessarily the easiest, but it allows us to cover some ground that we won't get to do in lecture.<br />
<br />
2a. Using the coset action, show that the normal core of H is the kernel of a homomorphism f from SL_2(Z/pZ) to S_3.<br />
<br />
2b. A ''unipotent matrix'' is a matrix A such that (A-I)^n = 0 for some positive integer n. Prove that a unipotent element of SL_2(Z/pZ) has order dividing p.<br />
<br />
2c. You may use the fact that unipotent matrices generate SL_2(Z/pZ). Given this fact, show that f must be trivial when p > 3.<br />
<br />
2d. Using the above, show that SL_2(Z/pZ) has no index-3 subgroups when p > 3.<br />
<br />
2e. (extra credit) What can you say about the minimal index of a proper subgroup of SL_2(Z/pZ)?<br />
<br />
3. Let H be a subgroup of G of index 2. Prove that H is normal.<br />
<br />
4. We proved two theorems in class: Lagrange's theorem, which says that the order of a subgroup of G divides |G|, and Cauchy's theorem, which says that if p divides |G| then there exists a subgroup of G of order p. Cauchy's theorem is a kind of partial converse to Lagrange's theorem, but the full converse is false. Give an example of a finite group G and a divisor n of |G| such that you can prove there is no subgroup of G of order n.<br />
<br />
5. Let X be the set of lines in F_3^2 (here a line means a 1-dimensional linear subspace) so that |X| = 4. Let G be the projective linear group PGL_2(F_3), so that G acts on X via its action on F_3^2. This action can be thought of as a homomorphism from G to S_4. Show that this homomorphism is an ISOMORPHISM from PGL_2(F_3) to S_4.<br />
<br />
6. The group S_3 has only three conjugacy classes. Prove that a finite group with at most three conjugacy classes has order at most 6.<br />
<br />
7. Let X be the set of ordered triples of elements of {1,..,n}, for some n >= 3. Then S_n acts on X. How many orbits are there? (Hint: the answer does not depend on n.)<br />
<br />
8. Suppose that G and H are groups and f: G -> H is a homomorphism. Recall the definition of the abelianization G^ab from the previous problem set.<br />
<br />
8a. Show that the composition G -> H -> H^ab factors through a unique homomorphism G^ab -> H^ab, which we denote f^ab.<br />
<br />
8b. Show that if f: G -> H and g: H -> Q are homomorphisms, then f^ab o g^ab = (f o g)^ab.<br />
<br />
(For those reading MacLane, this constitutes a proof that abelianization is a '''functor''' from the category of groups to the category of abelian groups.)<br />
<br />
==HOMEWORK 5 (due Oct 23)==<br />
<br />
1. Show that there are only two different isomorphism classes of groups of order 14. (Hint: prove that such a group must be a semidirect product of a group of order 7 by a group of order 2.)<br />
<br />
2. (variant of Burnside's Lemma) Let G be a finite group and H a normal subgroup. Suppose G acts on a finite set X. <br />
<br />
2a. Show that if x and y lie in the same H-orbit, and g is an element of G, then gx and gy also lie in the same H-orbit. Thus, G/H has a natural action on the set of H-orbits X/H.<br />
<br />
2b. Show that the average number of fixed points in X of an element of the coset gH is the same as the number of fixed points of the element gH of G/H in its action on X/H.<br />
<br />
3. A ''central extension'' of a group G by an abelian group A is a group E, together with a surjective homomorphism E -> G whose kernel is central in E and is isomorphic to A.<br />
<br />
3a. Show that the quaternion group is a central extension of (Z/2Z)^2 by Z/2Z.<br />
<br />
3b. Give an example of another nonabelian group which is a central extension of (Z/2Z)^2 by Z/2Z and which is not isomorphic to the quaternion group.<br />
<br />
4. Prove that a central extension of an abelian group is nilpotent.<br />
<br />
5. Give two different composition series for S_4 and show that they have the same composition factors.<br />
<br />
6. Suppose that A is a finitely generated abelian group, and B is a subgroup of A.<br />
<br />
6a. Prove that B is finitely generated and that rank(B) <= rank(A).<br />
<br />
6b. Prove that rank(A/B) = rank(A) - rank(B).<br />
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7. For this problem, you may use the fact (which I should have asked on homework earlier) that if H_1 and H_2 are subgroups of G, then<br />
<br />
[G: H_1 intersect H_2] <= [G:H_1][G:H_2].<br />
<br />
Prove that if G is a simple group, d is an integer greater than 1, and S is a subgroup of index d in G, then |G| <= d^d.<br />
<br />
Conclude that, in particular, A_5 has no subgroup of index 3. On the other hand, exhibit a subgroup of A_5 of index 5.<br />
<br />
(Note: d^d is not a sharp bound; can you get a bound of d factorial instead?)<br />
<br />
<!--(This argument shows that a solvable subgroup of A_n has size at most about (n-1)!, but this is far from sharp; e.g. I found a citation to a 1967 paper of Dixon which shows that a solvable subgroup of A_n has size at most k^n where k is around 2.88.)--><br />
<br />
<!--<br />
8. Give an example of an abelian group A with a subgroup B such that B is not a direct summand of A.<br />
<br />
9. Let A be the subgroup of Z^2 generated by the vector (1,0). Let B be the subgroup of Z^2 generated by the vector (3,3). Show that A and B are isomorphic to each other, but Z^2 / A is not isomorphic to Z^2 / B.<br />
<br />
==HOMEWORK 6 (due Oct 30)==<br />
<br />
1. In this problem, you will construct a functor F from the category of finite sets to the category of complex vector spaces. I'll tell you what it does on objects: if X is a finite set, then F(X) is the vector space spanned by a set of basis elements {e_x}_{x in X}, so that dim F(X) = |X|. Your job: complete the definition of F by describing the map from Mor_{Finite Sets}(X,Y) to Mor_{Vector Spaces}(F(X),F(Y)), and showing that your map respects composition.<br />
<br />
2. If (V, rho) is a representation of a group G, denote by V^G the space of vectors in V such that gv = v for all g in G; these are called _invariant_ vectors. We saw that the permutation representation of S_3 has a one-dimensional space of invariants. Prove that if X is a finite set with G-action, and V_X the corresponding permutation representation of G, then<br />
<br />
dim V_X^G = number of orbits of X.<br />
<br />
3. (Invariants are functorial) Let G be a group. Then there is a category called C[G]-Mod whose objects are complex representations of G (i.e. pairs (V,rho) where V is a complex vector space and rho is a homomorphism from G to GL(V)). The morphisms in this category are the ones described in class: Mor((V,rho),(W,psi)) is the set of linear maps f from V to W such that<br />
<br />
f(rho(g)(v)) = psi(g)(f(v))<br />
<br />
for all g in G and all v in V.<br />
<br />
Show that there is a functor H_0 from the category of representations of G to the category of vector spaces whose action on objects is given by<br />
<br />
H_0((V) = V^G.<br />
<br />
(In other words: your job again is to explain how to associate to a morphism from (V,rho) to (W,phi) a morphism from V^G to W^G, in a way that's compatible with composition.)<br />
<br />
<!--4. The group PSL_2(Z) has a presentation of the form [U,V|U^2 = V^3 = 1]. Thus, an n-dimensional representation of PSL_2(Z) is precisely a pair of n x n matrices (rho(U),rho(V)), such that rho(U)^2 = rho(V)^3 = 1. Prove that for every n > 1 there are uncountably many isomorphism classes of n-dimensional representations of PSL_2(Z). (Extra credit: show that there are uncountably many isomorphism classes of _irreducible_ representations.)<br />
<br />
An amazing theorem of Margulis (superrigidity) shows that SL_2(Z) is very special among arithmetic groups in this way. For instance, there are only _finitely_ many isomorphism classes of n-dimensional representations of PSL_d(Z) for any d > 2; those groups are "rigid."--><br />
<br />
<!--<br />
5. Let G be a finite group, let X be a set with G-action, and let chi be the character of the corresponding permutation representation. Show that, for each g in G, chi(g) is the number of elements of X fixed by g. (I am actually going to prove this in class.)<br />
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6. Let V be the space of homogeneous degree-3 polynomials in variables x_1, x_2, x_3; then S_3 acts on V by permutation of the three variables. Describe the decomposition of V into irreducible representations of S_3.<br />
<br />
7a. Show that if chi is the character of a representation of a finite group G, then chi(g^{-1}) is the complex conjugate of chi.<br />
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7b. As a consequence, show that if g is conjugate to g^{-1} for all g in G, then chi(g) is real.<br />
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7c. The groups D_p and S_n have the property that g is conjugate to g^{-1} for _every_ g in G, so that every character is a real-valued function on G. Give an example of a group G where this is not the case, and give an example of a representation of G whose character takes non-real values.<br />
<br />
==HOMEWORK 7 (due Nov 6)==<br />
<br />
1. Let G be a finite group, let X be a finite set with G-action, and let chi be the character of the corresponding permutation representation. Show that, for each g in G, chi(g) is the number of elements of X fixed by g. (I am actually going to prove this in class.)<br />
<br />
2a. Show that if chi is the character of a finite-dimensional representation of a finite group G, then chi(g^{-1}) is the complex conjugate of chi(g) for all g.<br />
<br />
2b. As a consequence, show that if g is conjugate to g^{-1} for all g in G, then chi(g) is real.<br />
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2c. The groups D_p and S_n have the property that g is conjugate to g^{-1} for _every_ g in G, so that every character is a real-valued function on G. Give an example of a group G where this is not the case, and give an example of a representation of G whose character takes non-real values.<br />
<br />
3. Let V and W be complex vector spaces. Let f be a linear transformation of V, and g a linear transformation of W. <br />
<br />
3a. Show that there is a unique linear transformation F satisfying<br />
<br />
F(v tensor w) = f(v) tensor g(w)<br />
<br />
for all v in V and all w in W. We denote this transformation by f tensor g.<br />
<br />
3b. Suppose V and W are finite dimensional. Show that Trace(f tensor g) = Trace(f) Trace(g). What does this say when f and g are both the identity transformation?<br />
<br />
4. Let V be a vector space. We define Sym^2 V to be the quotient of V tensor V by the subspace generated by all elements of the form<br />
<br />
v tensor w - w tensor v<br />
<br />
for v,w in V.<br />
<br />
Suppose dim V = n. What is dim Sym^2 V?<br />
<br />
5. Show that Z/pZ tensor Z/qZ is zero when p and q are distinct primes.<br />
<br />
==HOMEWORK 8 (due Nov 13)==<br />
<br />
1. (Schur's lemma can be false over non-algebraically closed fields) Let G be Z/3Z and let g be a generator of G. Then G acts by cyclic permutations on the set {1,2,3}; let P be the corresponding permutation representation on the REAL (not complex!) vector space generated by e_1,e_2,e_3.<br />
<br />
1a. Show that P breaks up as the direct sum of the trivial representation and a 2-dimensional representation V which is irreducible. (Remember, we are not working over the complex numbers, so you cannot use the criterion for irreducibility in terms of the character that we worked out in class for the complex case; you need to prove directly that V has no nontrivial proper subrepresentation.)<br />
<br />
1b. Show that the map from V to V induced by g is G-equivariant, but is neither zero nor a scalar multiplication.<br />
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2. If V_1 and V_2 are representations (not necessarily irreducible!) of G, show that <chi_V_1, chi V_2> is a non-negative integer.<br />
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3. When a formula always yields a non-negative integer, it is often because it is secretly a formula for either the cardinality of some finite set or the dimension of some finite-dimensional vector space. In this case, show that<br />
<br />
<chi_{V_1}, chi_{V_2}> = dim_C Hom_G(V_1,V_2)<br />
<br />
where we recall that Hom_G(V_1,G_2) refers to the space of G-equivariant homomorphisms from V_1 to V_2.<br />
<br />
4. If A is an abelian group, show that every irreducible representation of A is 1-dimensional. (Hint: show that there is a vector which is an eigenvector for every element of A!)<br />
<br />
5. Let G be the group of order 20 which is the semidirect product of A by H, where A is Z/5Z and H is (Z/5Z)^* with its natural action on A. We will work out all the irreducible representations of A. <br />
<br />
5a. First of all, consider the 5-dimensional permutation representation of G acting by left multiplication on the cosets of H. Show that this representation decomposes as a direct sum of the trivial representation and an irreducible 4-dimensional representation. (Hint: it is probably easier to use the character of this representation than to prove irreducibility directly.) <br />
<br />
5b. Now show that there are 4 1-dimensional representations of G which factor through the quotient G/A (i.e. such that rho(A) is the identity.)<br />
<br />
5c. Using the decomposition of the regular representation, show that the irreducible representations of G you have constructed are the only ones.<br />
<br />
==HOMEWORK 9 (due Nov 20)==<br />
<br />
0. Find all ring homomorphisms from Z to Z/30Z. [Warning: there are 8 of them, not 30.]<br />
<br />
<br />
1. If R is a ring (not necessarily with 1) such that a^2 = a for every a in R, R is called a Boolean ring.<br />
<br />
1a. Show that any nonzero Boolean ring has characteristic 2. (In other words, show that a+a=0 for all a in R.)<br />
<br />
1b. Show that every Boolean ring is commutative.<br />
<br />
<br />
2. Let phi : R -> S be a homomorphism of commutative rings with 1.<br />
<br />
2a. If I is an ideal of S, show that phi^(-1) (I) is an ideal of R.<br />
<br />
2b. If P is a prime ideal of S, show that phi^(-1) (P) is either R or a prime ideal of R.<br />
<br />
2c. If M is a maximal ideal of S and phi is surjective, show that phi^(-1) (M) is a maximal ideal of R.<br />
<br />
2d. Find a homomorphism phi : R -> S and a maximal ideal M of S such that phi^(-1) (M) is not a maximal ideal of R.<br />
<br />
<br />
3. Let Nil(A) denote the set of nilpotent elements of a ring A, and let R be a commutative ring. In class it was observed that Nil(R) is an ideal of R.<br />
<br />
3a. Find (a generator of) Nil(Z/720Z).<br />
<br />
3b. Show that Nil(R/Nil(R)) = 0.<br />
<br />
3c. If x is in Nil(R) and u is a unit, show that u+x is a unit.<br />
<br />
3d. Show that Nil(S), where S is the 2x2 matrices with real coefficients, is not an ideal of S.<br />
<br />
<br />
4a. If M is an R-module and I is an ideal of R such that im = 0 for all i in I and m in M, show that M carries the structure of an R/I-module.<br />
<br />
4b. If S is a commutative ring with 1 and J is a maximal ideal of S, show that J/J^2 is a vector space over S/J.<br />
<br />
<br />
5. A "divisible element" x of a Z-module is one such that for each positive integer k, there exists a y such that ky=x. Show that a free Z-module cannot contain a nonzero divisible element.<br />
<br />
<br />
6. Prove that the direct sum of any collection of free R-modules is also a free R-module.<br />
<br />
<br />
7. In this problem you will prove that the module M given by the direct product of copies of Z, indexed by the positive integers, is not a free Z-module. Therefore, assume that M is a free Z-module with basis B, and let N be the direct sum of copies of Z, as a submodule of M.<br />
<br />
7a. Show there is a countable subset B1 of B such that N is contained in the submodule N1 generated by B1. [Hint: N is countable.]<br />
<br />
7b. Show that M/N1 is a free Z-module.<br />
<br />
7c. Let S = {(b1, b2, b3, ...) : b_i = i! or -i!}. Show that there is some s in S which is not in N1. [Hint: S is uncountable.]<br />
<br />
7d. For any s in S but not N1, show that for each positive integer k, there exists some m in M with s-km in N1. Conclude that s+N1 is a divisible element of M/N1.<br />
<br />
7e. Use problem 5 to obtain a contradiction.<br />
<br />
<br />
8. In class I said "rank" is not always well-defined for R-modules if R is not commutative. Here is an example demonstrating this assertion. Let M be the countably infinite direct product of copies of Z as in exercise 7, and let R be the endomorphism ring of M. Define phi_1(a1,a2,a3,a4,...) = (a1,a3,a5,...) and phi_2(a1,a2,a3,a4,...) = (a2,a4,a6,...).<br />
<br />
8a. Show that {phi_1, phi_2} is a free basis for R, as a left R-module. [Hint: let psi_1(a1,a2,a3,...) = (a1,0,a2,0,...) and psi_2(a1,a2,a3,...) = (0,a1,0,a2,...). Compute the products of the phi_i and psi_j in either order, and use the result to show that the phi_i are a basis.]<br />
<br />
8b. Show that R is isomorphic to R^n for every positive integer n.<br />
<br />
<br />
==HOMEWORK 10 (due Dec 4)==<br />
<br />
1. '''How algebraists do calculus'''. The ''ring of dual numbers'' is the ring R = C[e]/(e^2). We think of e as an infinitesimal number; that is, something so small that its square is 0. This algebraic notion of infinitesimal is the basis for the algebraic notion of differentiation, which works as follows. Let f be a polynomial in R[x]. Show that there is a unique polynomial g in C[x] such that<br />
<br />
f(x+e) - f(x) = eg(x)<br />
<br />
and that in fact g is the derivative of f. In other words, the usual definition of "derivative" works just fine in this context, without any use of the notion of limit!<br />
<br />
2. Write down all the ideals of C[e]/e^2. (There are three.)<br />
<br />
3. Show that any right ideal of the ring M_2(Q) of 2x2 matrices is an ideal of the form I_V described in class for some subspace V of Q^2; that is, it consists precisely of those linear transformations whose image is contained in V.<br />
<br />
<br />
4. An ''idempotent'' element of a ring is an element e satisfying e^2 = e.<br />
<br />
4a. If e is an idempotent, show that 1-e is also an idempotent.<br />
<br />
4b. Suppose R has a unit (we are actually always supposing this, but I just want to remind you of it for this problem) and suppose e is a ''central'' idempotent, i.e. an idempotent which commutes with all elements of R. Show that eR and (1-e)R are both two-sided ideals. Moreover, show that eR and (1-e)R are also subrings of R (with the warning that the identity in eR is not 1, which is not contained in eR, but rather e.)<br />
<br />
4c. Show that the map f_e: R -> eR defined by f_e(r) = er is a ring homomorphism. Finally show that the ring homomorphism<br />
<br />
f_{e} x f_{1-e}: R -> eR x (1-e)R <br />
<br />
is actually an isomorphism of rings. In other words, the presence of a central idempotent induces a decomposition a ring as a direct product.<br />
<br />
<br />
5. The ''center'' of a ring R is the subring of elements commuting with every element of R.<br />
<br />
5a. Show that the center of M_n(Q) is the ring of scalar matrices.<br />
<br />
5b. Show that the center of the group ring C[G] is the set of elements sum f(g) g such that f(g) is a class function.<br />
<br />
<br />
6. '''How algebraists do differential equations.''' We define the ''Weyl algebra'' W of differential operators in one variable as follows. Multiplication by x is a linear transformation on the complex vector space V of all smooth functions on C. Differentiation, which we denote d, is also a linear transformation on V. So we let W be the subring of End(V) generated by x and d. We note that x and d satisfy the relation xd = dx - 1; in particular, the Weyl algebra is not commutative.<br />
<br />
6a. Describe the submodule of V which is annihilated by the element d-x in W.<br />
<br />
6b. Show that, for any polynomial f(x), fd-df = -f', and also show that for any polynomial g(d), gx-xg = -g'. (This is another way algebraists do calculus!) <br />
<br />
6c. Use the facts from (b) to show that the center of the Weyl algebra is just the field of constants. (Remark: you will use in the course of this proof the fact that a polynomial whose derivative is 0 is constant -- this would not be true if we were working in positive characteristic, as the example of x^p considered as a polynomial over F_p demonstrates.)<br />
<br />
<br />
7. If M is a left module for a ring R, we denote by Ann(M) the set of elements r in R such that rm = 0 for all m in M. Show that Ann(M) is a two-sided ideal. <br />
<br />
<br />
8. Let M be a left module for R, and let m be an element of M. We denote by Ann(m) the set of elements r in R such that rm = 0.<br />
<br />
8a. Show that Ann(m) is a left ideal, and show that the submodule Rm of M generated by m is isomorphic to the left module R/Ann(m).<br />
<br />
8c. Let V,W be as in question 6. Show that Ann(e^x) is the left ideal W(d-1).<br />
<br />
==HOMEWORK 11 (due Dec 13)==<br />
<br />
1. Let f: R -> S be a homomorphism of rings. As discussed in class, if M is a left R-module, then S tensor_R M is a left S-module, and if N is an S-module, then N can also be considered as an R-module by pullback of the action.<br />
<br />
Prove that if M is a finitely generated left R-module, then S tensor_R M is a finitely generated left S-module. On the other hand, give an example to show that N can be finitely generated as a left S-module but not finitely generated when considered as a left R-module.<br />
<br />
2. Recall from class that when V and W are complex vector spaces, W^dual tensor V is naturally isomorphic to the space of linear homomorphisms Hom(W,V).<br />
<br />
Now suppose that V and W are left C[G]-modules, i.e. complex representations of a group G. Show that W^dual tensor_C[G} V is the maximal G-invariant quotient of Hom(W,V); in other words, the quotient of Hom(W,V) by the subspace generated by all functions F of the form F(w) = f(gw) - gf(w) for some function f in Hom(W,V) and some g in G.<br />
<br />
3. '''How algebraists do logic.''' Why are Boolean rings called Boolean rings? Because they model some features of the Boolean arithmetic that governs propositional logic. Let A be a collection of propositions. We define two operations on A as follows. By a + b we mean the negation of a xor b, while by ab we mean "a or b."<br />
<br />
3a. It is not a priori obvious that these operations satisfy the ring axioms, but assume this for the moment. Show that a^2 = a for all a (so A is a boolean ring) and 2a = 0 for all a (so A has characteristic 2.)<br />
<br />
3b. Suppose we have a consistent assignment of truth-values to the propositions in A. Show that the set of true statements of A forms an ideal of A.<br />
<br />
3c. (Optional for people who like propositional logic) Show that a(b+c) = ab + ac with the given operations.<br />
<br />
<br />
4. Let K be Q[x]/(x^3-2).<br />
<br />
4a. Show that K is a field.<br />
<br />
4b. Show that K tensor_Q K decomposes as the direct sum of two fields. What are they? [Note: K tensor_Q K is a ring!]<br />
<br />
5. There was some discussion in class about when we can "go back and forth" between right modules and left modules, as is the case with group rings.<br />
<br />
5a. If R is a ring, we denote R^op ("the opposite ring") to be the ring whose elements are those of R, but whose multiplication law x_opp is defined by s x_opp r = rs. Show that if M is a left R-module, then the map M x R^op -> M sending (x,r) to rx makes M into a right R^op module. In particular, if there is a RING ISOMORPHISM from R to R^op, then every left R-module can be thought of as a right R-module.<br />
<br />
5b. Verify that the map sending g to g^{-1} extends by linearity to give a ring isomorphism between C[G] and C[G]^op.<br />
<br />
5c. Construct a ring isomorphism between M_n(Q) and M_n(Q)^op.<br />
<br />
5d. Optional: Give an example of a ring (with 1) which is not isomorphic to its opposite ring. (This is not so easy, I think! I found a MathOverflow thread with several examples, none of which came with a short proof of the non-isomorphism -- try your hand!)<br />
<br />
6. . Let R be a commutative ring with 1. If R has a unique maximal ideal M, R is called a "local ring".<br />
<br />
6a. If R is a local ring, show that every element of R that is not in M is a unit.<br />
<br />
6b. Conversely, if R is such that the set of nonunits forms an ideal I, show that R is a local ring with maximal ideal I.<br />
<br />
7. The inclusion of S_3 into S_4 associated to the inclusion of {1,2,3} into {1,2,3,4} induces a ring homomorphism from C[S_3} to C{S_4]. Let V be a representation of C{S_3]. Then, as discussed in class, V tensor_C[S_3] C[S_4] is a representation of S_4, called the '''induced representation''' of V.<br />
<br />
7a. Show that dim_C (V tensor_C[S_3] C[S_4]) = 4 dim_C V.<br />
<br />
7b. When V is the trivial representation of S_3, decompose (V tensor_C[S_3] C[S_4]) as a sum of irreducible representations of S_4.<br />
<br />
--><br />
<br />
<!--==HOMEWORK PROBLEMS TO GIVE IN FUTURE SEMESTERS==<br />
<br />
Let f: C[x] tensor_C C[y] -> C[z] be the map of rings with f(x) = f(y) = z. Show that f(m tensor n) is not 0 for any pure tensor m tensor n, but f is not injective. (This is a warning not to try to check injectivity by checking that no pure tensors are killed by the map.)<br />
<br />
--></div>Dummithttps://hilbert.math.wisc.edu/wiki/index.php?title=742&diff=53327422013-05-09T02:25:54Z<p>Dummit: /* HOMEWORK 11 (due May 10) */</p>
<hr />
<div>'''Math 742'''<br />
<br />
Commutative Algebra and Galois Theory<br />
<br />
Prof: [http://www.math.wisc.edu/~andreic Andrei Caldararu]<br />
<br />
Grader: [http://www.math.wisc.edu/~dummit/index.html Evan Dummit]<br />
<br />
Caldararu's office hours: Monday 1:30pm.<br />
<br />
Grader's office hours: Wednesday 2:15pm. Late homework may be given directly to the grader, along with either (i) the instructor's permission, or (ii) a polite request for mercy.<br />
<br />
This course, the second semester of the introductory graduate sequence in algebra, will cover the basic aspects of commutative ring theory and Galois theory. The textbook we'll use for the Commutative Algebra portion will be Atiyah-Macdonald "Commutative Algebra". For Galois Theory I plan on using Emil Artin's notes which are available [http://projecteuclid.org/DPubS?service=UI&version=1.0&verb=Display&handle=euclid.ndml/1175197041 here,] but I may change my mind before we start on it.<br />
<br />
"Fields and Galois Theory" by J.S. Milne is located [http://www.jmilne.org/math/CourseNotes/ft.html here].<br />
<br />
==SYLLABUS==<br />
<br />
In this space we will record the theorems and definitions we covered each week, which we can use as a list of notions you should be prepared to answer questions about on the Algebra qualifying exam. The material covered on the homework is also an excellent guide to the scope of the course. <br />
<br />
'''WEEKS 0.5-2''': <br />
<br />
Commutative rings and homomorphisms, integral domains, fields. Ideals, prime and maximal. Existence of maximal ideals. Local rings. Some geometric pictures (Spec and Specm). Nilradical. The nilradical is the intersection of all primes. Relatively prime ideals, Chinese remainder theorem. Extension and contraction, pictures of what happens for the inclusion Z -> Z[i]<br />
<br />
'''WEEK 3''':<br />
<br />
Modules, module homomorphisms. Module Hom, covariance and contravariance. Finitely generated modules, Nakayama's lemma and variants. Short exact sequences. Tensor product and exactness properties.<br />
<br />
'''WEEK 4''':<br />
<br />
An A-B-bimodule M induces a functor Mod-A -> Mod-B given by -- \otimes_A M. The equivalence of categories Mod-k and Mod-M_n(k). Restriction and extension of scalars. Algebras. Tensor product of algebras, the case of C \otimes_R C. Exactness of tensor products, flatness.<br />
<br />
'''WEEK 5'''<br />
<br />
Simple rings. Structure of finite dimensional simple rings over a field. Brauer group, computation of Br(R) = Z/2Z. Semisimple modules and rings. Wedderburn and Artin-Wedderburn theorems. Maschke's theorem on semisimplicity of k[G] for a finite ring. Tensor product of algebras and interpretation as fiber product of affine schemes. Localization (definition and basic properties).<br />
<br />
'''WEEK 6'''<br />
<br />
Rings and modules of fractions. Definitions and universal properties. Examples A_f, A_p. Modules of fractions. The operation S^{-1} is exact, corresponds to tensoring with S^{-1}A, so S^{-1}A is flat. Local properties, examples. Ideals in rings of fractions, in particular primes in A_p are primes in A which are contained in p.<br />
<br />
'''WEEK 7'''<br />
<br />
Integral dependence and valuations. Various characterizations of integral dependence, integral closure forms a ring, which is integrally closed. Relations to rings of fractions. Going up and going down. Valuation rings definition and basic properties. Existence. The integral closure of A is the intersection of all valuation rings which contain A. Nullstellensatz.<br />
<br />
Below you will find a repository of homework problems.<br />
<br />
==HOMEWORK 1 (due Feb 4)==<br />
<br />
Atiyah-Macdonald, page 10: 1, 2, 6, 10, 12, 15, 16, 17, 18, 21<br />
<br />
==HOMEWORK 2 (due Feb 15)==<br />
<br />
Atiyah-Macdonald, page 10: 19, 22, 26, 27, 28; page 31: 2, 3, 4, 8, 9<br />
<br />
==HOMEWORK 3 (due Feb 22)==<br />
<br />
Atiyah-Macdonald, page 31: 10, 11 (second part is hard!), 12, 13, 14, 20, 24<br />
<br />
==HOMEWORK 4 (due Mar 1)==<br />
<br />
Some non-commutative algebra exercises. You may find some references to read [http://stacks.math.columbia.edu/download/brauer.pdf here]. Or, even better, you can go and read Chapter I of the excellent book "Noncommutative Algebra" by Benson Farb and R. Keith Dennis. (On reserve in the library.) Most of these exercises are from this book.<br />
<br />
1) Let A, A' be k-algebras, with subalgebras B, B', respectively. If the centralizers of B, B' are C, C' (in A, A', resp.), then the centralizer of B\otimes_k B' in A\otimes_k A' is C\otimes C'.<br />
<br />
2) Find all the two-sided ideals of M_n(R), where R is a ring. Conclude that if D is a division algebra, M_n(D) is simple. (Hint: all two-sided ideals are of the form M_n(I) for some two-sided ideal I of R.)<br />
<br />
3) Let N be a submodule of the R-module M. If N and M/N are semisimple, does it follow that M is semisimple?<br />
<br />
4) Let M be a module such that every submodule is a direct summand. Show that M is semisimple as follows:<br />
<br />
(a) Show that every submodule of M inherits the property that every submodule is a direct summand.<br />
<br />
(b) Show that M contains a simple submodule: choose any finitely generated non-zero submodule M' of M. Let M" be a maximal submodule of M' such that M" is not equal to M' (why does it exist?). Then M'/M" is simple.<br />
<br />
(c) Let M_1 be the submodule of M generated by all simple submodules. Show that M_1 = M.<br />
<br />
5) Let A be a simple k-algebra with center k such that [A:k] = p^2 for a prime number p. Prove that A is either a division algebra or A is M_p(k).<br />
<br />
==HOMEWORK 5 (due Mar 8)==<br />
<br />
Atiyah-Macdonald page 31: 5, 6, 10; page 43: 1, 3, 5, 12, 13, 14<br />
<br />
==HOMEWORK 6 (due Mar 15)==<br />
<br />
Atiyah-Macdonald page 43: 15, 18, 19, 20, 21, 22, 25; page 67: 3, 9<br />
<br />
==HOMEWORK 7 (due Mar 22)==<br />
<br />
Atiyah-Macdonald page 67: 1, 2, 17, 28, 30, 31.<br />
<br />
Also prove that the integral closure of R = k[x,y]/(y^2-x^3) is isomorphic to k[t] (we discussed this in class), as follows. Consider the map R -> k[t] given by x|->t^2, y|->t^3. Show that it is injective, so we can consider R as a subring of k[t]. Moreover, they have the same field of fractions. Now on one hand, k[t] is integrally closed, so the closure of R must be included in it. On the other hand, t is integral over R, so k[t] is contained in the integral closure of R.<br />
<br />
==HOMEWORK 8 (due Apr 12)==<br />
<br />
Atiyah-Macdonald page 84: 2, 4, 5, 7, 14, 26, 27<br />
<br />
Milne page 24: 1.1--1.4<br />
<br />
==HOMEWORK 9 (due Apr 19)==<br />
<br />
Milne page 31: 2.1--2.6<br />
<br />
==HOMEWORK 10 (due May 3)==<br />
<br />
1. ''Standard Facts about Finite Fields''<br />
<br />
Let p be a prime. Observe that if f(x) is an irreducible polynomial of degree d over F_p (the field with p elements), then (F_p)[x]/(f(x)) is a field with p^d elements, which we call F_(p^d).<br />
<br />
a. Show that x^(p^d) - x splits into a product of distinct linear factors over F_(p^d) by showing that every element of F_(p^d) is a root of this polynomial.<br />
<br />
b. Show that the splitting field of x^(p^d) - x is F_(p^d). Conclude that the field with p^d elements is unique up to isomorphism (i.e., calling it "the field" is justified).<br />
<br />
c. Let sigma be the Frobenius automorphism a->a^p. Prove that the Galois group of F_(p^d)/F_p is cyclic of degree d and is generated by sigma. Conclude that the subfields of F_(p^d) are the fields F_(p^k) where k divides d.<br />
<br />
d. Prove that x^(p^d) - x factors over F_p as the product of all the monic irreducible polynomials over F_p whose degree divides d. Use this to find the number of irreducible cubic polynomials over F_7.<br />
<br />
[Note: as all of these are "standard facts" you can likely look all of them up. Do this only after you have tried to prove them from scratch.]<br />
<br />
<br />
2. Let k be a field, f be a polynomial of degree n in k[x], and K be the splitting field of f over k.<br />
<br />
a. Show that [K:k] divides n!.<br />
<br />
b. Show that in order to have [K:k] = n!, it is necessary but not sufficient for f to be irreducible.<br />
<br />
<br />
3. Find the Galois group of x^7 - 2 explicitly as a permutation group on the roots.<br />
<br />
<br />
4. a. Show that the splitting field K of x^8-2 is Q(2^(1/8), zeta_8) where zeta_8 is a primitive eighth root of unity.<br />
<br />
b. Despite the facts that [Q(2^(1/8)) : Q] = 8 and [Q(zeta_8) : Q] = 4, prove that [K:Q] is actually 16, not 32. (Optional: also explain how you will avoid making similar mistakes in the future, if you have made them in the past.)<br />
<br />
c. Find generators for Gal(K/Q) and write explicitly their permutation action on the roots of x^8-2.<br />
<br />
<br />
5. Let E=k(alpha) where alpha is algebraic over k.<br />
<br />
a. If [E:k] is odd, prove that k(alpha^2) = k(alpha).<br />
<br />
b. Show more generally that k(alpha^2) = k(alpha) if and only if the minimal polynomial for alpha has an odd-degree term. (In other words, if it has a term b*x^c where b is nonzero and c is odd.)<br />
<br />
c. Is it true in general that if m and n are relatively prime and alpha is algebraic of degree m over k, then k(alpha) = k(alpha^n) ?<br />
<br />
<br />
6. Find the splitting fields and Galois groups of the following polynomials (if they exist):<br />
<br />
a. x^3 - 3 over Q.<br />
<br />
b. x^3 - x + 1 over Q.<br />
<br />
c. x^3 - 3 over Q(sqrt 3).<br />
<br />
d. x^3 - 3 over Q(sqrt(-3)).<br />
<br />
e. x^4 - 2 over Q.<br />
<br />
f. x^4 - 7 over Q.<br />
<br />
==HOMEWORK 11 (due May 10)==<br />
<br />
<br />
1. a. Find the splitting field K of x^4 - 4x^2 - 1 over Q.<br />
<br />
b. Show that Gal(K/Q) is (isomorphic to) the dihedral group of order 8.<br />
<br />
c. Find the 8 nontrivial subfields of K and say which 4 of them are Galois over Q.<br />
<br />
<br />
2. Prove or disprove: Every field of degree 4 over Q has a subfield of degree 2 over Q.<br />
<br />
<br />
3. Let K/F be an algebraic extension. We say that an element alpha in K is "abelian" if F[alpha] is a Galois extension of F and the Galois group Gal(F[alpha]/F) is abelian. Prove that the set of abelian elements of K is a field.<br />
<br />
<br />
4. a. Let F < K < L be a tower of field extensions with [L:F] finite, and let alpha be an element of L. If p(x) is the minimal polynomial of alpha over F, prove that K tensor_F F(alpha) is isomorphic to K[x]/p(x) as a K-algebra.<br />
<br />
b. Let K1 and K2 be two algebraic extensions of a field K contained in a field L of characteristic zero. Prove that the K-algebra K1 tensor_K K2 has no nonzero nilpotent elements.</div>Dummithttps://hilbert.math.wisc.edu/wiki/index.php?title=742&diff=53287422013-05-06T19:32:03Z<p>Dummit: /* HOMEWORK 11 (due May 10) */</p>
<hr />
<div>'''Math 742'''<br />
<br />
Commutative Algebra and Galois Theory<br />
<br />
Prof: [http://www.math.wisc.edu/~andreic Andrei Caldararu]<br />
<br />
Grader: [http://www.math.wisc.edu/~dummit/index.html Evan Dummit]<br />
<br />
Caldararu's office hours: Monday 1:30pm.<br />
<br />
Grader's office hours: Wednesday 2:15pm. Late homework may be given directly to the grader, along with either (i) the instructor's permission, or (ii) a polite request for mercy.<br />
<br />
This course, the second semester of the introductory graduate sequence in algebra, will cover the basic aspects of commutative ring theory and Galois theory. The textbook we'll use for the Commutative Algebra portion will be Atiyah-Macdonald "Commutative Algebra". For Galois Theory I plan on using Emil Artin's notes which are available [http://projecteuclid.org/DPubS?service=UI&version=1.0&verb=Display&handle=euclid.ndml/1175197041 here,] but I may change my mind before we start on it.<br />
<br />
"Fields and Galois Theory" by J.S. Milne is located [http://www.jmilne.org/math/CourseNotes/ft.html here].<br />
<br />
==SYLLABUS==<br />
<br />
In this space we will record the theorems and definitions we covered each week, which we can use as a list of notions you should be prepared to answer questions about on the Algebra qualifying exam. The material covered on the homework is also an excellent guide to the scope of the course. <br />
<br />
'''WEEKS 0.5-2''': <br />
<br />
Commutative rings and homomorphisms, integral domains, fields. Ideals, prime and maximal. Existence of maximal ideals. Local rings. Some geometric pictures (Spec and Specm). Nilradical. The nilradical is the intersection of all primes. Relatively prime ideals, Chinese remainder theorem. Extension and contraction, pictures of what happens for the inclusion Z -> Z[i]<br />
<br />
'''WEEK 3''':<br />
<br />
Modules, module homomorphisms. Module Hom, covariance and contravariance. Finitely generated modules, Nakayama's lemma and variants. Short exact sequences. Tensor product and exactness properties.<br />
<br />
'''WEEK 4''':<br />
<br />
An A-B-bimodule M induces a functor Mod-A -> Mod-B given by -- \otimes_A M. The equivalence of categories Mod-k and Mod-M_n(k). Restriction and extension of scalars. Algebras. Tensor product of algebras, the case of C \otimes_R C. Exactness of tensor products, flatness.<br />
<br />
'''WEEK 5'''<br />
<br />
Simple rings. Structure of finite dimensional simple rings over a field. Brauer group, computation of Br(R) = Z/2Z. Semisimple modules and rings. Wedderburn and Artin-Wedderburn theorems. Maschke's theorem on semisimplicity of k[G] for a finite ring. Tensor product of algebras and interpretation as fiber product of affine schemes. Localization (definition and basic properties).<br />
<br />
'''WEEK 6'''<br />
<br />
Rings and modules of fractions. Definitions and universal properties. Examples A_f, A_p. Modules of fractions. The operation S^{-1} is exact, corresponds to tensoring with S^{-1}A, so S^{-1}A is flat. Local properties, examples. Ideals in rings of fractions, in particular primes in A_p are primes in A which are contained in p.<br />
<br />
'''WEEK 7'''<br />
<br />
Integral dependence and valuations. Various characterizations of integral dependence, integral closure forms a ring, which is integrally closed. Relations to rings of fractions. Going up and going down. Valuation rings definition and basic properties. Existence. The integral closure of A is the intersection of all valuation rings which contain A. Nullstellensatz.<br />
<br />
Below you will find a repository of homework problems.<br />
<br />
==HOMEWORK 1 (due Feb 4)==<br />
<br />
Atiyah-Macdonald, page 10: 1, 2, 6, 10, 12, 15, 16, 17, 18, 21<br />
<br />
==HOMEWORK 2 (due Feb 15)==<br />
<br />
Atiyah-Macdonald, page 10: 19, 22, 26, 27, 28; page 31: 2, 3, 4, 8, 9<br />
<br />
==HOMEWORK 3 (due Feb 22)==<br />
<br />
Atiyah-Macdonald, page 31: 10, 11 (second part is hard!), 12, 13, 14, 20, 24<br />
<br />
==HOMEWORK 4 (due Mar 1)==<br />
<br />
Some non-commutative algebra exercises. You may find some references to read [http://stacks.math.columbia.edu/download/brauer.pdf here]. Or, even better, you can go and read Chapter I of the excellent book "Noncommutative Algebra" by Benson Farb and R. Keith Dennis. (On reserve in the library.) Most of these exercises are from this book.<br />
<br />
1) Let A, A' be k-algebras, with subalgebras B, B', respectively. If the centralizers of B, B' are C, C' (in A, A', resp.), then the centralizer of B\otimes_k B' in A\otimes_k A' is C\otimes C'.<br />
<br />
2) Find all the two-sided ideals of M_n(R), where R is a ring. Conclude that if D is a division algebra, M_n(D) is simple. (Hint: all two-sided ideals are of the form M_n(I) for some two-sided ideal I of R.)<br />
<br />
3) Let N be a submodule of the R-module M. If N and M/N are semisimple, does it follow that M is semisimple?<br />
<br />
4) Let M be a module such that every submodule is a direct summand. Show that M is semisimple as follows:<br />
<br />
(a) Show that every submodule of M inherits the property that every submodule is a direct summand.<br />
<br />
(b) Show that M contains a simple submodule: choose any finitely generated non-zero submodule M' of M. Let M" be a maximal submodule of M' such that M" is not equal to M' (why does it exist?). Then M'/M" is simple.<br />
<br />
(c) Let M_1 be the submodule of M generated by all simple submodules. Show that M_1 = M.<br />
<br />
5) Let A be a simple k-algebra with center k such that [A:k] = p^2 for a prime number p. Prove that A is either a division algebra or A is M_p(k).<br />
<br />
==HOMEWORK 5 (due Mar 8)==<br />
<br />
Atiyah-Macdonald page 31: 5, 6, 10; page 43: 1, 3, 5, 12, 13, 14<br />
<br />
==HOMEWORK 6 (due Mar 15)==<br />
<br />
Atiyah-Macdonald page 43: 15, 18, 19, 20, 21, 22, 25; page 67: 3, 9<br />
<br />
==HOMEWORK 7 (due Mar 22)==<br />
<br />
Atiyah-Macdonald page 67: 1, 2, 17, 28, 30, 31.<br />
<br />
Also prove that the integral closure of R = k[x,y]/(y^2-x^3) is isomorphic to k[t] (we discussed this in class), as follows. Consider the map R -> k[t] given by x|->t^2, y|->t^3. Show that it is injective, so we can consider R as a subring of k[t]. Moreover, they have the same field of fractions. Now on one hand, k[t] is integrally closed, so the closure of R must be included in it. On the other hand, t is integral over R, so k[t] is contained in the integral closure of R.<br />
<br />
==HOMEWORK 8 (due Apr 12)==<br />
<br />
Atiyah-Macdonald page 84: 2, 4, 5, 7, 14, 26, 27<br />
<br />
Milne page 24: 1.1--1.4<br />
<br />
==HOMEWORK 9 (due Apr 19)==<br />
<br />
Milne page 31: 2.1--2.6<br />
<br />
==HOMEWORK 10 (due May 3)==<br />
<br />
1. ''Standard Facts about Finite Fields''<br />
<br />
Let p be a prime. Observe that if f(x) is an irreducible polynomial of degree d over F_p (the field with p elements), then (F_p)[x]/(f(x)) is a field with p^d elements, which we call F_(p^d).<br />
<br />
a. Show that x^(p^d) - x splits into a product of distinct linear factors over F_(p^d) by showing that every element of F_(p^d) is a root of this polynomial.<br />
<br />
b. Show that the splitting field of x^(p^d) - x is F_(p^d). Conclude that the field with p^d elements is unique up to isomorphism (i.e., calling it "the field" is justified).<br />
<br />
c. Let sigma be the Frobenius automorphism a->a^p. Prove that the Galois group of F_(p^d)/F_p is cyclic of degree d and is generated by sigma. Conclude that the subfields of F_(p^d) are the fields F_(p^k) where k divides d.<br />
<br />
d. Prove that x^(p^d) - x factors over F_p as the product of all the monic irreducible polynomials over F_p whose degree divides d. Use this to find the number of irreducible cubic polynomials over F_7.<br />
<br />
[Note: as all of these are "standard facts" you can likely look all of them up. Do this only after you have tried to prove them from scratch.]<br />
<br />
<br />
2. Let k be a field, f be a polynomial of degree n in k[x], and K be the splitting field of f over k.<br />
<br />
a. Show that [K:k] divides n!.<br />
<br />
b. Show that in order to have [K:k] = n!, it is necessary but not sufficient for f to be irreducible.<br />
<br />
<br />
3. Find the Galois group of x^7 - 2 explicitly as a permutation group on the roots.<br />
<br />
<br />
4. a. Show that the splitting field K of x^8-2 is Q(2^(1/8), zeta_8) where zeta_8 is a primitive eighth root of unity.<br />
<br />
b. Despite the facts that [Q(2^(1/8)) : Q] = 8 and [Q(zeta_8) : Q] = 4, prove that [K:Q] is actually 16, not 32. (Optional: also explain how you will avoid making similar mistakes in the future, if you have made them in the past.)<br />
<br />
c. Find generators for Gal(K/Q) and write explicitly their permutation action on the roots of x^8-2.<br />
<br />
<br />
5. Let E=k(alpha) where alpha is algebraic over k.<br />
<br />
a. If [E:k] is odd, prove that k(alpha^2) = k(alpha).<br />
<br />
b. Show more generally that k(alpha^2) = k(alpha) if and only if the minimal polynomial for alpha has an odd-degree term. (In other words, if it has a term b*x^c where b is nonzero and c is odd.)<br />
<br />
c. Is it true in general that if m and n are relatively prime and alpha is algebraic of degree m over k, then k(alpha) = k(alpha^n) ?<br />
<br />
<br />
6. Find the splitting fields and Galois groups of the following polynomials (if they exist):<br />
<br />
a. x^3 - 3 over Q.<br />
<br />
b. x^3 - x + 1 over Q.<br />
<br />
c. x^3 - 3 over Q(sqrt 3).<br />
<br />
d. x^3 - 3 over Q(sqrt(-3)).<br />
<br />
e. x^4 - 2 over Q.<br />
<br />
f. x^4 - 7 over Q.<br />
<br />
==HOMEWORK 11 (due May 10)==<br />
<br />
<br />
1. a. Find the splitting field K of x^4 - 4x^2 - 1 over Q.<br />
<br />
b. Show that Gal(K/Q) is (isomorphic to) the dihedral group of order 8.<br />
<br />
c. Find the 8 nontrivial subfields of K and say which 4 of them are Galois over Q.<br />
<br />
<br />
2. Prove or disprove: Every field of degree 4 over Q has a subfield of degree 2 over Q.<br />
<br />
<br />
3. Let K/F be an algebraic extension. We say that an element alpha in K is "abelian" if F[alpha] is a Galois extension of F and the Galois group Gal(F[alpha]/F) is abelian. Prove that the set of abelian elements of K is a field.<br />
<br />
<br />
4. a. Let F < K < L be a tower of field extensions with [L:F] finite, and let alpha be an element of L. If p(x) is the minimal polynomial of alpha over F, prove that K tensor_F F(alpha) is isomorphic to K[x]/p(x) as a K-algebra.<br />
<br />
b. Let K1 and K2 be two algebraic extensions of a field K contained in a field L of characteristic zero. Prove that the F-algebra K1 tensor_K K2 has no nonzero nilpotent elements.</div>Dummithttps://hilbert.math.wisc.edu/wiki/index.php?title=742&diff=53277422013-05-06T19:29:27Z<p>Dummit: /* HOMEWORK 11 (due May 10) */</p>
<hr />
<div>'''Math 742'''<br />
<br />
Commutative Algebra and Galois Theory<br />
<br />
Prof: [http://www.math.wisc.edu/~andreic Andrei Caldararu]<br />
<br />
Grader: [http://www.math.wisc.edu/~dummit/index.html Evan Dummit]<br />
<br />
Caldararu's office hours: Monday 1:30pm.<br />
<br />
Grader's office hours: Wednesday 2:15pm. Late homework may be given directly to the grader, along with either (i) the instructor's permission, or (ii) a polite request for mercy.<br />
<br />
This course, the second semester of the introductory graduate sequence in algebra, will cover the basic aspects of commutative ring theory and Galois theory. The textbook we'll use for the Commutative Algebra portion will be Atiyah-Macdonald "Commutative Algebra". For Galois Theory I plan on using Emil Artin's notes which are available [http://projecteuclid.org/DPubS?service=UI&version=1.0&verb=Display&handle=euclid.ndml/1175197041 here,] but I may change my mind before we start on it.<br />
<br />
"Fields and Galois Theory" by J.S. Milne is located [http://www.jmilne.org/math/CourseNotes/ft.html here].<br />
<br />
==SYLLABUS==<br />
<br />
In this space we will record the theorems and definitions we covered each week, which we can use as a list of notions you should be prepared to answer questions about on the Algebra qualifying exam. The material covered on the homework is also an excellent guide to the scope of the course. <br />
<br />
'''WEEKS 0.5-2''': <br />
<br />
Commutative rings and homomorphisms, integral domains, fields. Ideals, prime and maximal. Existence of maximal ideals. Local rings. Some geometric pictures (Spec and Specm). Nilradical. The nilradical is the intersection of all primes. Relatively prime ideals, Chinese remainder theorem. Extension and contraction, pictures of what happens for the inclusion Z -> Z[i]<br />
<br />
'''WEEK 3''':<br />
<br />
Modules, module homomorphisms. Module Hom, covariance and contravariance. Finitely generated modules, Nakayama's lemma and variants. Short exact sequences. Tensor product and exactness properties.<br />
<br />
'''WEEK 4''':<br />
<br />
An A-B-bimodule M induces a functor Mod-A -> Mod-B given by -- \otimes_A M. The equivalence of categories Mod-k and Mod-M_n(k). Restriction and extension of scalars. Algebras. Tensor product of algebras, the case of C \otimes_R C. Exactness of tensor products, flatness.<br />
<br />
'''WEEK 5'''<br />
<br />
Simple rings. Structure of finite dimensional simple rings over a field. Brauer group, computation of Br(R) = Z/2Z. Semisimple modules and rings. Wedderburn and Artin-Wedderburn theorems. Maschke's theorem on semisimplicity of k[G] for a finite ring. Tensor product of algebras and interpretation as fiber product of affine schemes. Localization (definition and basic properties).<br />
<br />
'''WEEK 6'''<br />
<br />
Rings and modules of fractions. Definitions and universal properties. Examples A_f, A_p. Modules of fractions. The operation S^{-1} is exact, corresponds to tensoring with S^{-1}A, so S^{-1}A is flat. Local properties, examples. Ideals in rings of fractions, in particular primes in A_p are primes in A which are contained in p.<br />
<br />
'''WEEK 7'''<br />
<br />
Integral dependence and valuations. Various characterizations of integral dependence, integral closure forms a ring, which is integrally closed. Relations to rings of fractions. Going up and going down. Valuation rings definition and basic properties. Existence. The integral closure of A is the intersection of all valuation rings which contain A. Nullstellensatz.<br />
<br />
Below you will find a repository of homework problems.<br />
<br />
==HOMEWORK 1 (due Feb 4)==<br />
<br />
Atiyah-Macdonald, page 10: 1, 2, 6, 10, 12, 15, 16, 17, 18, 21<br />
<br />
==HOMEWORK 2 (due Feb 15)==<br />
<br />
Atiyah-Macdonald, page 10: 19, 22, 26, 27, 28; page 31: 2, 3, 4, 8, 9<br />
<br />
==HOMEWORK 3 (due Feb 22)==<br />
<br />
Atiyah-Macdonald, page 31: 10, 11 (second part is hard!), 12, 13, 14, 20, 24<br />
<br />
==HOMEWORK 4 (due Mar 1)==<br />
<br />
Some non-commutative algebra exercises. You may find some references to read [http://stacks.math.columbia.edu/download/brauer.pdf here]. Or, even better, you can go and read Chapter I of the excellent book "Noncommutative Algebra" by Benson Farb and R. Keith Dennis. (On reserve in the library.) Most of these exercises are from this book.<br />
<br />
1) Let A, A' be k-algebras, with subalgebras B, B', respectively. If the centralizers of B, B' are C, C' (in A, A', resp.), then the centralizer of B\otimes_k B' in A\otimes_k A' is C\otimes C'.<br />
<br />
2) Find all the two-sided ideals of M_n(R), where R is a ring. Conclude that if D is a division algebra, M_n(D) is simple. (Hint: all two-sided ideals are of the form M_n(I) for some two-sided ideal I of R.)<br />
<br />
3) Let N be a submodule of the R-module M. If N and M/N are semisimple, does it follow that M is semisimple?<br />
<br />
4) Let M be a module such that every submodule is a direct summand. Show that M is semisimple as follows:<br />
<br />
(a) Show that every submodule of M inherits the property that every submodule is a direct summand.<br />
<br />
(b) Show that M contains a simple submodule: choose any finitely generated non-zero submodule M' of M. Let M" be a maximal submodule of M' such that M" is not equal to M' (why does it exist?). Then M'/M" is simple.<br />
<br />
(c) Let M_1 be the submodule of M generated by all simple submodules. Show that M_1 = M.<br />
<br />
5) Let A be a simple k-algebra with center k such that [A:k] = p^2 for a prime number p. Prove that A is either a division algebra or A is M_p(k).<br />
<br />
==HOMEWORK 5 (due Mar 8)==<br />
<br />
Atiyah-Macdonald page 31: 5, 6, 10; page 43: 1, 3, 5, 12, 13, 14<br />
<br />
==HOMEWORK 6 (due Mar 15)==<br />
<br />
Atiyah-Macdonald page 43: 15, 18, 19, 20, 21, 22, 25; page 67: 3, 9<br />
<br />
==HOMEWORK 7 (due Mar 22)==<br />
<br />
Atiyah-Macdonald page 67: 1, 2, 17, 28, 30, 31.<br />
<br />
Also prove that the integral closure of R = k[x,y]/(y^2-x^3) is isomorphic to k[t] (we discussed this in class), as follows. Consider the map R -> k[t] given by x|->t^2, y|->t^3. Show that it is injective, so we can consider R as a subring of k[t]. Moreover, they have the same field of fractions. Now on one hand, k[t] is integrally closed, so the closure of R must be included in it. On the other hand, t is integral over R, so k[t] is contained in the integral closure of R.<br />
<br />
==HOMEWORK 8 (due Apr 12)==<br />
<br />
Atiyah-Macdonald page 84: 2, 4, 5, 7, 14, 26, 27<br />
<br />
Milne page 24: 1.1--1.4<br />
<br />
==HOMEWORK 9 (due Apr 19)==<br />
<br />
Milne page 31: 2.1--2.6<br />
<br />
==HOMEWORK 10 (due May 3)==<br />
<br />
1. ''Standard Facts about Finite Fields''<br />
<br />
Let p be a prime. Observe that if f(x) is an irreducible polynomial of degree d over F_p (the field with p elements), then (F_p)[x]/(f(x)) is a field with p^d elements, which we call F_(p^d).<br />
<br />
a. Show that x^(p^d) - x splits into a product of distinct linear factors over F_(p^d) by showing that every element of F_(p^d) is a root of this polynomial.<br />
<br />
b. Show that the splitting field of x^(p^d) - x is F_(p^d). Conclude that the field with p^d elements is unique up to isomorphism (i.e., calling it "the field" is justified).<br />
<br />
c. Let sigma be the Frobenius automorphism a->a^p. Prove that the Galois group of F_(p^d)/F_p is cyclic of degree d and is generated by sigma. Conclude that the subfields of F_(p^d) are the fields F_(p^k) where k divides d.<br />
<br />
d. Prove that x^(p^d) - x factors over F_p as the product of all the monic irreducible polynomials over F_p whose degree divides d. Use this to find the number of irreducible cubic polynomials over F_7.<br />
<br />
[Note: as all of these are "standard facts" you can likely look all of them up. Do this only after you have tried to prove them from scratch.]<br />
<br />
<br />
2. Let k be a field, f be a polynomial of degree n in k[x], and K be the splitting field of f over k.<br />
<br />
a. Show that [K:k] divides n!.<br />
<br />
b. Show that in order to have [K:k] = n!, it is necessary but not sufficient for f to be irreducible.<br />
<br />
<br />
3. Find the Galois group of x^7 - 2 explicitly as a permutation group on the roots.<br />
<br />
<br />
4. a. Show that the splitting field K of x^8-2 is Q(2^(1/8), zeta_8) where zeta_8 is a primitive eighth root of unity.<br />
<br />
b. Despite the facts that [Q(2^(1/8)) : Q] = 8 and [Q(zeta_8) : Q] = 4, prove that [K:Q] is actually 16, not 32. (Optional: also explain how you will avoid making similar mistakes in the future, if you have made them in the past.)<br />
<br />
c. Find generators for Gal(K/Q) and write explicitly their permutation action on the roots of x^8-2.<br />
<br />
<br />
5. Let E=k(alpha) where alpha is algebraic over k.<br />
<br />
a. If [E:k] is odd, prove that k(alpha^2) = k(alpha).<br />
<br />
b. Show more generally that k(alpha^2) = k(alpha) if and only if the minimal polynomial for alpha has an odd-degree term. (In other words, if it has a term b*x^c where b is nonzero and c is odd.)<br />
<br />
c. Is it true in general that if m and n are relatively prime and alpha is algebraic of degree m over k, then k(alpha) = k(alpha^n) ?<br />
<br />
<br />
6. Find the splitting fields and Galois groups of the following polynomials (if they exist):<br />
<br />
a. x^3 - 3 over Q.<br />
<br />
b. x^3 - x + 1 over Q.<br />
<br />
c. x^3 - 3 over Q(sqrt 3).<br />
<br />
d. x^3 - 3 over Q(sqrt(-3)).<br />
<br />
e. x^4 - 2 over Q.<br />
<br />
f. x^4 - 7 over Q.<br />
<br />
==HOMEWORK 11 (due May 10)==<br />
<br />
<br />
1. a. Find the splitting field K of x^4 - 4x^2 - 1 over Q.<br />
<br />
b. Show that Gal(K/Q) is (isomorphic to) the dihedral group of order 8.<br />
<br />
c. Find the 8 nontrivial subfields of K and say which 4 of them are Galois.<br />
<br />
<br />
2. Prove or disprove: Every field of degree 4 over Q has a subfield of degree 2 over Q.<br />
<br />
<br />
3. Let K/F be an algebraic extension. We say that an element alpha in K is "abelian" if F[alpha] is a Galois extension of F and the Galois group Gal(F[alpha]/F) is abelian. Prove that the set of abelian elements of K is a field.<br />
<br />
<br />
4. a. Let F < K < L be a tower of field extensions with [L:F] finite, and let alpha be an element of L. If p(x) is the minimal polynomial of alpha over F, prove that K tensor_F F(alpha) is isomorphic to K[x]/p(x) as a K-algebra.<br />
<br />
b. Let K1 and K2 be two algebraic extensions of a field K contained in a field L of characteristic zero. Prove that the F-algebra K1 tensor_K K2 has no nonzero nilpotent elements.</div>Dummithttps://hilbert.math.wisc.edu/wiki/index.php?title=742&diff=50617422013-02-15T03:08:15Z<p>Dummit: </p>
<hr />
<div>'''Math 742'''<br />
<br />
Commutative Algebra and Galois Theory<br />
<br />
Prof: [http://www.math.wisc.edu/~andreic Andrei Caldararu]<br />
<br />
Grader: [http://www.math.wisc.edu/~dummit/index.html Evan Dummit]<br />
<br />
Caldararu's office hours: Monday 1:30pm.<br />
<br />
Grader's office hours: Wednesday 2:15pm. Late homework may be given directly to the grader, along with either (i) the instructor's permission, or (ii) a polite request for mercy.<br />
<br />
This course, the second semester of the introductory graduate sequence in algebra, will cover the basic aspects of commutative ring theory and Galois theory. The textbook we'll use for the Commutative Algebra portion will be Atiyah-Macdonald "Commutative Algebra". For Galois Theory I plan on using Emil Artin's notes which are available [http://projecteuclid.org/DPubS?service=UI&version=1.0&verb=Display&handle=euclid.ndml/1175197041 here,] but I may change my mind before we start on it.<br />
<br />
==SYLLABUS==<br />
<br />
In this space we will record the theorems and definitions we covered each week, which we can use as a list of notions you should be prepared to answer questions about on the Algebra qualifying exam. The material covered on the homework is also an excellent guide to the scope of the course. <br />
<br />
'''WEEK 1''': <br />
<br />
Commutative rings and homomorphisms, integral domains, fields. Ideals, prime and maximal. Existence of maximal ideals. Local rings. Some geometric pictures (Spec and Specm). Nilradical. The nilradical is the intersection of all primes. Relatively prime ideals, Chinese remainder theorem. Extension and contraction, pictures of what happens for the inclusion Z -> Z[i]<br />
<br />
'''WEEK 2''':<br />
<br />
Modules, module homomorphisms. Module Hom, covariance and contravariance. Finitely generated modules, Nakayama's lemma and variants. Short exact sequences. Tensor product and exactness properties.<br />
<br />
<br />
<br />
Below you will find a repository of homework problems.<br />
<br />
==HOMEWORK 1 (due Feb 4)==<br />
<br />
Atiyah-Macdonald, page 10: 1, 2, 6, 10, 12, 15, 16, 17, 18, 21<br />
<br />
==HOMEWORK 2 (due Feb 15)==<br />
<br />
Atiyah-Macdonald, page 10: 19, 22, 26, 27, 28; page 31: 2, 3, 4, 8, 9</div>Dummithttps://hilbert.math.wisc.edu/wiki/index.php?title=742&diff=49727422013-01-29T03:40:24Z<p>Dummit: </p>
<hr />
<div>'''Math 742'''<br />
<br />
Commutative Algebra and Galois Theory<br />
<br />
Prof: [http://www.math.wisc.edu/~andreic Andrei Caldararu]<br />
<br />
Grader: [http://www.math.wisc.edu/~dummit/personal.html Evan Dummit]<br />
<br />
Caldararu's office hours: Monday 1:30pm.<br />
<br />
Grader's office hours: Wednesday 2:15pm. Late homework may be given directly to the grader, along with either (i) the instructor's permission, or (ii) a polite request for mercy.<br />
<br />
This course, the second semester of the introductory graduate sequence in algebra, will cover the basic aspects of commutative ring theory and Galois theory. The textbook we'll use for the Commutative Algebra portion will be Atiyah-Macdonald "Commutative Algebra". For Galois Theory I plan on using Emil Artin's notes which are available [http://projecteuclid.org/DPubS?service=UI&version=1.0&verb=Display&handle=euclid.ndml/1175197041 here,] but I may change my mind before we start on it.<br />
<br />
==SYLLABUS==<br />
<br />
In this space we will record the theorems and definitions we covered each week, which we can use as a list of notions you should be prepared to answer questions about on the Algebra qualifying exam. The material covered on the homework is also an excellent guide to the scope of the course. <br />
<br />
'''WEEK 1''': <br />
<br />
Commutative rings and homomorphisms, integral domains, fields. Ideals, prime and maximal. Existence of maximal ideals. Local rings. Some geometric pictures (Spec and Specm). Nilradical.<br />
<br />
<br />
Below you will find a repository of homework problems. <br />
<br />
==HOMEWORK 1 (due Feb 4)==<br />
<br />
Atiyah-Macdonald, page 10: 1, 2, 6, 10, 12, 15, 16, 17, 18, 21</div>Dummithttps://hilbert.math.wisc.edu/wiki/index.php?title=741&diff=48137412012-12-22T10:36:29Z<p>Dummit: /* HOMEWORK 11 (due Dec 13) */</p>
<hr />
<div>'''Math 741'''<br />
<br />
Algebra<br />
<br />
Prof: [http://www.math.wisc.edu/~ellenber Jordan Ellenberg]<br />
<br />
Grader: Evan Dummit<br />
<br />
Ellenberg's office hours: Friday 3pm.<br />
<br />
Grader's office hours: Monday 4pm. Late homework may be given directly to the grader, along with either (i) the instructor's permission, or (ii) a polite request for mercy.<br />
<br />
This course, the first semester of the introductory graduate sequence in algebra, will cover the basic theory of groups, group actions, representation, linear and multilinear algebra, and the beginnings of ring theory.<br />
<br />
==SYLLABUS==<br />
<br />
In this space we will record the theorems and definitions we covered each week, which we can use as a list of notions you should be prepared to answer questions about on the Algebra qualifying exam. The material covered on the homework is also an excellent guide to the scope of the course. <br />
<br />
'''WEEK 1''': <br />
<br />
Definition of group. Associativity. Inverse. <br />
<br />
Examples of group: GL_n(R). GL_n(Z). Z/nZ. R. Z. R^*. The free group F_k on k generators. <br />
<br />
Homomorphisms. The homomorphisms from F_k to G are in bijection with G^k. Isomorphisms.<br />
<br />
'''WEEK 2''':<br />
<br />
The symmetric group (or permutation group) S_n on n letters. Cycle decomposition of a permutation. Order of a permutation. Thm: every element of a finite group has finite order. <br />
<br />
Subgroups. Left and right cosets. Lagrange's Theorem. Cyclic groups. The order of an element of a finite group is a divisor of the order of the group.<br />
<br />
The sign homomorphism S_n -> +-1.<br />
<br />
<br />
'''WEEK 3'''<br />
<br />
Normal subgroups. The quotient of a group by a normal subgroup. The first isomorphism theorem. Examples of S_n -> +-1 and S_4 -> S_3 with kernel V_4, the Klein 4-group.<br />
<br />
Centralizers and centers. Abelian groups. The center of SL_n(R) is either 1 or +-1.<br />
<br />
Groups with presentations. The infinite dihedral group <x,y | x^2 = 1, y^2 = 1>.<br />
<br />
'''WEEK 4'''<br />
<br />
More on groups with presentations.<br />
<br />
Second and third isomorphism theorems.<br />
<br />
Semidirect products.<br />
<br />
'''WEEK 5'''<br />
<br />
Group actions, orbits, and stabilizers.<br />
<br />
Orbit-stabilizer theorem.<br />
<br />
Cayley's theorem.<br />
<br />
Cauchy's theorem.<br />
<br />
'''WEEK 6'''<br />
<br />
Applications of orbit-stabilizer theorem (p-groups have nontrivial center, first Sylow theorem.)<br />
<br />
Classification of finite abelian groups and finitely generated abelian groups.<br />
<br />
Composition series and the Jordan-Holder theorem (which we state but don't prove.) <br />
<br />
The difference between knowing the composition factors and knowing the group (e.g. all p-groups of the same order have the same composition factors.)<br />
<br />
'''WEEK 7'''<br />
<br />
Simplicity of A_n.<br />
<br />
Nilpotent groups (main example: the Heisenberg group)<br />
<br />
Derived series and lower central series.<br />
<br />
Category theory 101: Definition of category and functor. Some examples. A group is a groupoid with one object.<br />
<br />
'''WEEK 8'''<br />
<br />
Introduction to representation theory.<br />
<br />
'''WEEK 10'''<br />
<br />
Ring theory 101: Rings, ring homomorphisms, ideals, isomorphism theorems. Examples: fields, Z, the Hamilton quaternions, matrix rings, rings of polynomials and formal power series, quadratic integer rings, group rings. Integral domains. Maximal and prime ideals. The nilradical.<br />
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Module theory 101: Modules, module homomorphisms, submodules, isomorphism theorems. Noetherian modules and Zorn's lemma. Direct sums and direct products of arbitrary collections of modules.<br />
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Below you will find a repository of homework problems. Note that some of these problems are taken from Lang's ''Algebra''.<br />
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==HOMEWORK 1 (due Sep 20)==<br />
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1. Suppose that H_1 and H_2 are subgroups of a group G. Prove that the intersection of H_1 and H_2 is a subgroup of G.<br />
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2. Recall that S_3 (the symmetric group) is the group of permutations of the set {1..3}. List all the subgroups of S_3.<br />
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3. We can define an equivalence relation on rational numbers by declaring two rational numbers to be equal whenever they differ by an integer. We denote the set of equivalence classes by Q/Z. The operation of addition makes Q/Z into a group.<br />
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a) For each n, prove that Q/Z has a subgroup of order n.<br />
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b) Prove that Q/Z is a ''divisible'' group: that is, if x is an element of Q/Z and n is an integer, there exists an element y of Q/Z such that ny = x. (Note that we write the operation in this group as addition rather than multiplication, which is why we write ny for the n-fold product of y with itself rather than y^n)<br />
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c) Prove that Q/Z is not finitely generated. (Hint: prove that if x_1, .. x_d is a finite subset of Q/Z, the subgroup of Q/Z generated by x_1, ... x_d is finite.)<br />
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4. Let F_2 be the free group on two generators x,y. Prove that there is a automorphism of F_2 which sends x to xyx and y to xy, and prove that this automorphism is unique.<br />
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5. Let H be a subgroup of G, and let N_G(H), the ''normalizer'' of H in G, be the set of elements g in G satsifying g H g^{-1} = H. Prove that N_G(H) is a subgroup of G.<br />
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6. Let T be the subgroup of GL_2(R) consisting of diagonal matrices. This is an example of a "Cartan subgroup," or "torus" (whence the notation T.) Describe the normalizer N(T) of T in GL_2(R). Prove that there is a homomorphism from N(T) to Z/2Z whose kernel is precisely T. <br />
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7. Prove that the group of inner automorphisms of a group is normal in the full automorphism group Aut(G).<br />
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8. Let H and H' be subgroups of a group G, and let x be an element of G. We denote by HxH' the set of elements of the form hxh', with h in H and h' in H', and we call HxH a ''double coset'' of the pair (H,H').<br />
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a) Show that G decomposes as a disjoint union of double cosets of (H,H').<br />
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b) Let G be GL_2(R) and let B be the subgroup of upper triangular matrices. Prove that G decomposes into exactly two double cosets of (B,B).<br />
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c) Let G be the symmetric group S_n, and let H be the subgroup of permutations which fix 1. Describe the double coset decomposition of S_n into double cosets of (H,H).<br />
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==HOMEWORK 2 (due Sep 27)==<br />
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1. Prove that the symmetric group S_n is generated by transpositions. Hint: proceed by induction on n. Let H be the subgroup of S_n consisting of permutations fixing 1. Observe that H is isomorphic to S_{n-1}, so that H is generated by transpositions. Explain why it suffices to show that, for any g in S_n, the coset gH contains a transposition. Then explain why this is the case.<br />
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2. Given that the symmetric group S_n is generated by transpositions, show that there are no nontrivial homomorphisms from S_n to Z/pZ for any odd prime p.<br />
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3. Compute the centralizer of the triple flip (12)(34)(56) in S_6. (Note: this is the kind of problem that a computer algebra package like sage is very good at, and I encourage you to learn to use sage well enough to check your answer.)<br />
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4. Warning: a normal subgroup of a normal subgroup need not be normal! Let H be the subgroup of S_4 generated by (12)(34). Then H is a subgroup of the Klein 4-group V_4, which we have already shown is normal in S_4. Show that H is normal in V_4, but H is not normal in S_4.<br />
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5. The '''quaternion group''' Q is a non-abelian finite group of order 8. Its elements are 1, -1, i, -i, j, -j, k, -k; it satisfies g^2 = -1 for all g except +-1, and ijk = 1. (Note that the notation "-i" means "the product of i with -1" and -1 is understood to be central in Q.) Prove that every subgroup of Q is normal.<br />
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6. (finishing example done in class) Let Gamma be the group F<x,y> / (x^2 = 1, y^2 = 1). (Remember, this means the quotient of the free group by the group generated by all conjugates of x^2 and y^2.) Prove that Gamma is infinite. Prove that Gamma has a subgroup R generated by xy isomorphic to Z, that R is normal, and that the quotient Gamma/R is isomorphic to Z/2Z. <br />
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7. Suppose a group G has the property that every element in g satisfies g^2 = 1. Show that G must be abelian.<br />
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==HOMEWORK 3 (due Oct 4)==<br />
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1. Multiplication by -1 is an automorphism of Z/nZ. Let a be the homomorphism Z/2Z -> Aut(Z/nZ) which sends the nontrivial element of Z/2Z to multiplication by -1. Then we have a semidirect product of Z/pZ by Z/2Z as defined in class. This group of order 2n is called a ''dihedral group.'' and is denoted D_n (or sometimes D_{2n}).<br />
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1a. Compute the center of D_n. (Note that the answer depends on n!)<br />
1b. Show that all involutions of D_n are conjugate to each other if and only if n is odd.<br />
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2. (More on direct products that we didn't do in class.) Suppose that G is a semidirect product of N with H. Suppose furthermore that H is also normal in G. Prove that G is a direct product of N with H.<br />
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3. Let Q be the quaternion group from last week's homework. Show that Q does not decompose as a semidirect product N \rtimes H, apart form the trivial cases N = {e} and N = Q.<br />
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4. The ''affine linear group'' of degree n is the group of transformations from R^n to R^n of the form x -> Ax + b, where A is a matrix in GL_n(R) and b is a vector in R^n. Prove that the affine linear group is a semidirect product R^n \rtimes GL_n.<br />
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5. The ''ordinary triangle group'' T(p,q,r) is the group with presentation <x,y | x^p = y^q = (xy)^r = 1>. This is a very interesting family of groups which arises naturally in many contexts in number theory and topology.<br />
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6a. Prove that T(2,2,n) is isomorphic to the dihedral group D_n.<br />
6b. Prove that T(2,3,3) is finite and compute its order. Is it isomorphic to a finite group we've discussed in class?<br />
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(It is a beautiful fact, but outside the scope of this course, that T(p,q,r) is finite if and only if 1/p + 1/q + 1/r > 1.)<br />
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7. Let G be a subgroup of GL_n(R) which contains SL_n(R). Prove that G is a normal subgroup of GL_n(R).<br />
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8. Let G be a group and let H_1 and H_2 be subgroups of G of finite index (that is, the number of cosets in G/H_i is finite, i = 1,2.) Prove that H_1 intersect H_2 also has finite index in G.<br />
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9. Recall that the commutator [x,y] is x y x^{-1} y^{-1}. If G is a group, the commutator subgroup of G, denoted G' or [G,G], is the subgroup of G generated by all commutators [x,y] for x,y in G.<br />
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9a. Show that G' is a normal subgroup of G.<br />
9b. Show that G/G' is an abelian group.<br />
9c. Show that if f: G -> A is a homomorphism, with A abelian, then G' is contained in ker f. Conclude that f factors through a homomorphism G/G' -> A.<br />
9d. Show that G' = G if and only if G has no nontrivial abelian quotient. In this case, we say G is ''perfect''.<br />
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10. Under what conditions on (p,q,r) is the triangle group T(p,q,r) perfect?<br />
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==HOMEWORK 4 (due Oct 16)==<br />
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1. Let X be a set with a transitive action of a group G, which we think of as a homomorphism f: G -> Sym(X). Let H_x be the stabilizer of an element x of X.<br />
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1a. If x' is another element of X, show that H_{x"} and H_x are conjugate subgroups. (Hint: this is not true without the hypothesis of transitivity, so make sure to use it!)<br />
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1b. Show that the kernel of f is the intersection of all the conjugates of H_x in G. (This subgroup is called the ''normal core'' of H_x.)<br />
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2. This week, a certified mathematical grown-up asked on MathOverflow what the index-3 subgroups of SL_2(Z/pZ) were. Let's do it ourselves for homework. Let p be a prime, and suppose H is an index-3 subgroup of SL_2(Z/pZ). I do not claim that the proof sketched below is necessarily the easiest, but it allows us to cover some ground that we won't get to do in lecture.<br />
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2a. Using the coset action, show that the normal core of H is the kernel of a homomorphism f from SL_2(Z/pZ) to S_3.<br />
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2b. A ''unipotent matrix'' is a matrix A such that (A-I)^n = 0 for some positive integer n. Prove that a unipotent element of SL_2(Z/pZ) has order dividing p.<br />
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2c. You may use the fact that unipotent matrices generate SL_2(Z/pZ). Given this fact, show that f must be trivial when p > 3.<br />
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2d. Using the above, show that SL_2(Z/pZ) has no index-3 subgroups when p > 3.<br />
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2e. (extra credit) What can you say about the minimal index of a proper subgroup of SL_2(Z/pZ)?<br />
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3. Let H be a subgroup of G of index 2. Prove that H is normal.<br />
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4. We proved two theorems in class: Lagrange's theorem, which says that the order of a subgroup of G divides |G|, and Cauchy's theorem, which says that if p divides |G| then there exists a subgroup of G of order p. Cauchy's theorem is a kind of partial converse to Lagrange's theorem, but the full converse is false. Give an example of a finite group G and a divisor n of |G| such that you can prove there is no subgroup of G of order n.<br />
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5. Let X be the set of lines in F_3^2 (here a line means a 1-dimensional linear subspace) so that |X| = 4. Let G be the projective linear group PGL_2(F_3), so that G acts on X via its action on F_3^2. This action can be thought of as a homomorphism from G to S_4. Show that this homomorphism is an ISOMORPHISM from PGL_2(F_3) to S_4.<br />
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6. The group S_3 has only three conjugacy classes. Prove that a finite group with at most three conjugacy classes has order at most 6.<br />
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7. Let X be the set of ordered triples of elements of {1,..,n}, for some n >= 3. Then S_n acts on X. How many orbits are there? (Hint: the answer does not depend on n.)<br />
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8. Suppose that G and H are groups and f: G -> H is a homomorphism. Recall the definition of the abelianization G^ab from the previous problem set.<br />
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8a. Show that the composition G -> H -> H^ab factors through a unique homomorphism G^ab -> H^ab, which we denote f^ab.<br />
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8b. Show that if f: G -> H and g: H -> Q are homomorphisms, then f^ab o g^ab = (f o g)^ab.<br />
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(For those reading MacLane, this constitutes a proof that abelianization is a '''functor''' from the category of groups to the category of abelian groups.)<br />
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==HOMEWORK 5 (due Oct 23)==<br />
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1. Show that there are only two different isomorphism classes of groups of order 14. (Hint: prove that such a group must be a semidirect product of a group of order 7 by a group of order 2.)<br />
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2. (variant of Burnside's Lemma) Let G be a finite group and H a normal subgroup. Suppose G acts on a finite set X. <br />
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2a. Show that if x and y lie in the same H-orbit, and g is an element of G, then gx and gy also lie in the same H-orbit. Thus, G/H has a natural action on the set of H-orbits X/H.<br />
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2b. Show that the average number of fixed points in X of an element of the coset gH is the same as the number of fixed points of the element gH of G/H in its action on X/H.<br />
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3. A ''central extension'' of a group G by an abelian group A is a group E, together with a surjective homomorphism E -> G whose kernel is central in E and is isomorphic to A.<br />
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3a. Show that the quaternion group is a central extension of (Z/2Z)^2 by Z/2Z.<br />
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3b. Give an example of another nonabelian group which is a central extension of (Z/2Z)^2 by Z/2Z and which is not isomorphic to the quaternion group.<br />
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4. Prove that a central extension of an abelian group is nilpotent.<br />
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5. Give two different composition series for S_4 and show that they have the same composition factors.<br />
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6. Suppose that A is a finitely generated abelian group, and B is a subgroup of A.<br />
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6a. Prove that B is finitely generated and that rank(B) <= rank(A).<br />
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6b. Prove that rank(A/B) = rank(A) - rank(B).<br />
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7. For this problem, you may use the fact (which I should have asked on homework earlier) that if H_1 and H_2 are subgroups of G, then<br />
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[G: H_1 intersect H_2] <= [G:H_1][G:H_2].<br />
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Prove that if G is a simple group, d is an integer greater than 1, and S is a subgroup of index d in G, then |G| <= d^d.<br />
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Conclude that, in particular, A_5 has no subgroup of index 3. On the other hand, exhibit a subgroup of A_5 of index 5.<br />
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(Note: d^d is not a sharp bound; can you get a bound of d factorial instead?)<br />
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<!--(This argument shows that a solvable subgroup of A_n has size at most about (n-1)!, but this is far from sharp; e.g. I found a citation to a 1967 paper of Dixon which shows that a solvable subgroup of A_n has size at most k^n where k is around 2.88.)--><br />
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8. Give an example of an abelian group A with a subgroup B such that B is not a direct summand of A.<br />
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9. Let A be the subgroup of Z^2 generated by the vector (1,0). Let B be the subgroup of Z^2 generated by the vector (3,3). Show that A and B are isomorphic to each other, but Z^2 / A is not isomorphic to Z^2 / B.<br />
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==HOMEWORK 6 (due Oct 30)==<br />
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1. In this problem, you will construct a functor F from the category of finite sets to the category of complex vector spaces. I'll tell you what it does on objects: if X is a finite set, then F(X) is the vector space spanned by a set of basis elements {e_x}_{x in X}, so that dim F(X) = |X|. Your job: complete the definition of F by describing the map from Mor_{Finite Sets}(X,Y) to Mor_{Vector Spaces}(F(X),F(Y)), and showing that your map respects composition.<br />
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2. If (V, rho) is a representation of a group G, denote by V^G the space of vectors in V such that gv = v for all g in G; these are called _invariant_ vectors. We saw that the permutation representation of S_3 has a one-dimensional space of invariants. Prove that if X is a finite set with G-action, and V_X the corresponding permutation representation of G, then<br />
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dim V_X^G = number of orbits of X.<br />
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3. (Invariants are functorial) Let G be a group. Then there is a category called C[G]-Mod whose objects are complex representations of G (i.e. pairs (V,rho) where V is a complex vector space and rho is a homomorphism from G to GL(V)). The morphisms in this category are the ones described in class: Mor((V,rho),(W,psi)) is the set of linear maps f from V to W such that<br />
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f(rho(g)(v)) = psi(g)(f(v))<br />
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for all g in G and all v in V.<br />
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Show that there is a functor H_0 from the category of representations of G to the category of vector spaces whose action on objects is given by<br />
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H_0((V) = V^G.<br />
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(In other words: your job again is to explain how to associate to a morphism from (V,rho) to (W,phi) a morphism from V^G to W^G, in a way that's compatible with composition.)<br />
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<!--4. The group PSL_2(Z) has a presentation of the form [U,V|U^2 = V^3 = 1]. Thus, an n-dimensional representation of PSL_2(Z) is precisely a pair of n x n matrices (rho(U),rho(V)), such that rho(U)^2 = rho(V)^3 = 1. Prove that for every n > 1 there are uncountably many isomorphism classes of n-dimensional representations of PSL_2(Z). (Extra credit: show that there are uncountably many isomorphism classes of _irreducible_ representations.)<br />
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An amazing theorem of Margulis (superrigidity) shows that SL_2(Z) is very special among arithmetic groups in this way. For instance, there are only _finitely_ many isomorphism classes of n-dimensional representations of PSL_d(Z) for any d > 2; those groups are "rigid."--><br />
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5. Let G be a finite group, let X be a set with G-action, and let chi be the character of the corresponding permutation representation. Show that, for each g in G, chi(g) is the number of elements of X fixed by g. (I am actually going to prove this in class.)<br />
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6. Let V be the space of homogeneous degree-3 polynomials in variables x_1, x_2, x_3; then S_3 acts on V by permutation of the three variables. Describe the decomposition of V into irreducible representations of S_3.<br />
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7a. Show that if chi is the character of a representation of a finite group G, then chi(g^{-1}) is the complex conjugate of chi.<br />
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7b. As a consequence, show that if g is conjugate to g^{-1} for all g in G, then chi(g) is real.<br />
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7c. The groups D_p and S_n have the property that g is conjugate to g^{-1} for _every_ g in G, so that every character is a real-valued function on G. Give an example of a group G where this is not the case, and give an example of a representation of G whose character takes non-real values.<br />
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==HOMEWORK 7 (due Nov 6)==<br />
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1. Let G be a finite group, let X be a finite set with G-action, and let chi be the character of the corresponding permutation representation. Show that, for each g in G, chi(g) is the number of elements of X fixed by g. (I am actually going to prove this in class.)<br />
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2a. Show that if chi is the character of a finite-dimensional representation of a finite group G, then chi(g^{-1}) is the complex conjugate of chi(g) for all g.<br />
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2b. As a consequence, show that if g is conjugate to g^{-1} for all g in G, then chi(g) is real.<br />
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2c. The groups D_p and S_n have the property that g is conjugate to g^{-1} for _every_ g in G, so that every character is a real-valued function on G. Give an example of a group G where this is not the case, and give an example of a representation of G whose character takes non-real values.<br />
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3. Let V and W be complex vector spaces. Let f be a linear transformation of V, and g a linear transformation of W. <br />
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3a. Show that there is a unique linear transformation F satisfying<br />
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F(v tensor w) = f(v) tensor g(w)<br />
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for all v in V and all w in W. We denote this transformation by f tensor g.<br />
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3b. Suppose V and W are finite dimensional. Show that Trace(f tensor g) = Trace(f) Trace(g). What does this say when f and g are both the identity transformation?<br />
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4. Let V be a vector space. We define Sym^2 V to be the quotient of V tensor V by the subspace generated by all elements of the form<br />
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v tensor w - w tensor v<br />
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for v,w in V.<br />
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Suppose dim V = n. What is dim Sym^2 V?<br />
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5. Show that Z/pZ tensor Z/qZ is zero when p and q are distinct primes.<br />
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==HOMEWORK 8 (due Nov 13)==<br />
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1. (Schur's lemma can be false over non-algebraically closed fields) Let G be Z/3Z and let g be a generator of G. Then G acts by cyclic permutations on the set {1,2,3}; let P be the corresponding permutation representation on the REAL (not complex!) vector space generated by e_1,e_2,e_3.<br />
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1a. Show that P breaks up as the direct sum of the trivial representation and a 2-dimensional representation V which is irreducible. (Remember, we are not working over the complex numbers, so you cannot use the criterion for irreducibility in terms of the character that we worked out in class for the complex case; you need to prove directly that V has no nontrivial proper subrepresentation.)<br />
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1b. Show that the map from V to V induced by g is G-equivariant, but is neither zero nor a scalar multiplication.<br />
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2. If V_1 and V_2 are representations (not necessarily irreducible!) of G, show that <chi_V_1, chi V_2> is a non-negative integer.<br />
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3. When a formula always yields a non-negative integer, it is often because it is secretly a formula for either the cardinality of some finite set or the dimension of some finite-dimensional vector space. In this case, show that<br />
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<chi_{V_1}, chi_{V_2}> = dim_C Hom_G(V_1,V_2)<br />
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where we recall that Hom_G(V_1,G_2) refers to the space of G-equivariant homomorphisms from V_1 to V_2.<br />
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4. If A is an abelian group, show that every irreducible representation of A is 1-dimensional. (Hint: show that there is a vector which is an eigenvector for every element of A!)<br />
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5. Let G be the group of order 20 which is the semidirect product of A by H, where A is Z/5Z and H is (Z/5Z)^* with its natural action on A. We will work out all the irreducible representations of A. <br />
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5a. First of all, consider the 5-dimensional permutation representation of G acting by left multiplication on the cosets of H. Show that this representation decomposes as a direct sum of the trivial representation and an irreducible 4-dimensional representation. (Hint: it is probably easier to use the character of this representation than to prove irreducibility directly.) <br />
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5b. Now show that there are 4 1-dimensional representations of G which factor through the quotient G/A (i.e. such that rho(A) is the identity.)<br />
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5c. Using the decomposition of the regular representation, show that the irreducible representations of G you have constructed are the only ones.<br />
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==HOMEWORK 9 (due Nov 20)==<br />
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0. Find all ring homomorphisms from Z to Z/30Z. [Warning: there are 8 of them, not 30.]<br />
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1. If R is a ring (not necessarily with 1) such that a^2 = a for every a in R, R is called a Boolean ring.<br />
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1a. Show that any nonzero Boolean ring has characteristic 2. (In other words, show that a+a=0 for all a in R.)<br />
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1b. Show that every Boolean ring is commutative.<br />
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2. Let phi : R -> S be a homomorphism of commutative rings with 1.<br />
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2a. If I is an ideal of S, show that phi^(-1) (I) is an ideal of R.<br />
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2b. If P is a prime ideal of S, show that phi^(-1) (P) is either R or a prime ideal of R.<br />
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2c. If M is a maximal ideal of S and phi is surjective, show that phi^(-1) (M) is a maximal ideal of R.<br />
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2d. Find a homomorphism phi : R -> S and a maximal ideal M of S such that phi^(-1) (M) is not a maximal ideal of R.<br />
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3. Let Nil(A) denote the set of nilpotent elements of a ring A, and let R be a commutative ring. In class it was observed that Nil(R) is an ideal of R.<br />
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3a. Find (a generator of) Nil(Z/720Z).<br />
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3b. Show that Nil(R/Nil(R)) = 0.<br />
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3c. If x is in Nil(R) and u is a unit, show that u+x is a unit.<br />
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3d. Show that Nil(S), where S is the 2x2 matrices with real coefficients, is not an ideal of S.<br />
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4a. If M is an R-module and I is an ideal of R such that im = 0 for all i in I and m in M, show that M carries the structure of an R/I-module.<br />
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4b. If S is a commutative ring with 1 and J is a maximal ideal of S, show that J/J^2 is a vector space over S/J.<br />
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5. A "divisible element" x of a Z-module is one such that for each positive integer k, there exists a y such that ky=x. Show that a free Z-module cannot contain a nonzero divisible element.<br />
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6. Prove that the direct sum of any collection of free R-modules is also a free R-module.<br />
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7. In this problem you will prove that the module M given by the direct product of copies of Z, indexed by the positive integers, is not a free Z-module. Therefore, assume that M is a free Z-module with basis B, and let N be the direct sum of copies of Z, as a submodule of M.<br />
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7a. Show there is a countable subset B1 of B such that N is contained in the submodule N1 generated by B1. [Hint: N is countable.]<br />
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7b. Show that M/N1 is a free Z-module.<br />
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7c. Let S = {(b1, b2, b3, ...) : b_i = i! or -i!}. Show that there is some s in S which is not in N1. [Hint: S is uncountable.]<br />
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7d. For any s in S but not N1, show that for each positive integer k, there exists some m in M with s-km in N1. Conclude that s+N1 is a divisible element of M/N1.<br />
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7e. Use problem 5 to obtain a contradiction.<br />
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8. In class I said "rank" is not always well-defined for R-modules if R is not commutative. Here is an example demonstrating this assertion. Let M be the countably infinite direct product of copies of Z as in exercise 7, and let R be the endomorphism ring of M. Define phi_1(a1,a2,a3,a4,...) = (a1,a3,a5,...) and phi_2(a1,a2,a3,a4,...) = (a2,a4,a6,...).<br />
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8a. Show that {phi_1, phi_2} is a free basis for R, as a left R-module. [Hint: let psi_1(a1,a2,a3,...) = (a1,0,a2,0,...) and psi_2(a1,a2,a3,...) = (0,a1,0,a2,...). Compute the products of the phi_i and psi_j in either order, and use the result to show that the phi_i are a basis.]<br />
<br />
8b. Show that R is isomorphic to R^n for every positive integer n.<br />
<br />
<br />
==HOMEWORK 10 (due Dec 4)==<br />
<br />
1. '''How algebraists do calculus'''. The ''ring of dual numbers'' is the ring R = C[e]/(e^2). We think of e as an infinitesimal number; that is, something so small that its square is 0. This algebraic notion of infinitesimal is the basis for the algebraic notion of differentiation, which works as follows. Let f be a polynomial in R[x]. Show that there is a unique polynomial g in C[x] such that<br />
<br />
f(x+e) - f(x) = eg(x)<br />
<br />
and that in fact g is the derivative of f. In other words, the usual definition of "derivative" works just fine in this context, without any use of the notion of limit!<br />
<br />
2. Write down all the ideals of C[e]/e^2. (There are three.)<br />
<br />
3. Show that any right ideal of the ring M_2(Q) of 2x2 matrices is an ideal of the form I_V described in class for some subspace V of Q^2; that is, it consists precisely of those linear transformations whose image is contained in V.<br />
<br />
<br />
4. An ''idempotent'' element of a ring is an element e satisfying e^2 = e.<br />
<br />
4a. If e is an idempotent, show that 1-e is also an idempotent.<br />
<br />
4b. Suppose R has a unit (we are actually always supposing this, but I just want to remind you of it for this problem) and suppose e is a ''central'' idempotent, i.e. an idempotent which commutes with all elements of R. Show that eR and (1-e)R are both two-sided ideals. Moreover, show that eR and (1-e)R are also subrings of R (with the warning that the identity in eR is not 1, which is not contained in eR, but rather e.)<br />
<br />
4c. Show that the map f_e: R -> eR defined by f_e(r) = er is a ring homomorphism. Finally show that the ring homomorphism<br />
<br />
f_{e} x f_{1-e}: R -> eR x (1-e)R <br />
<br />
is actually an isomorphism of rings. In other words, the presence of a central idempotent induces a decomposition a ring as a direct product.<br />
<br />
<br />
5. The ''center'' of a ring R is the subring of elements commuting with every element of R.<br />
<br />
5a. Show that the center of M_n(Q) is the ring of scalar matrices.<br />
<br />
5b. Show that the center of the group ring C[G] is the set of elements sum f(g) g such that f(g) is a class function.<br />
<br />
<br />
6. '''How algebraists do differential equations.''' We define the ''Weyl algebra'' W of differential operators in one variable as follows. Multiplication by x is a linear transformation on the complex vector space V of all smooth functions on C. Differentiation, which we denote d, is also a linear transformation on V. So we let W be the subring of End(V) generated by x and d. We note that x and d satisfy the relation xd = dx - 1; in particular, the Weyl algebra is not commutative.<br />
<br />
6a. Describe the submodule of V which is annihilated by the element d-x in W.<br />
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6b. Show that, for any polynomial f(x), fd-df = -f', and also show that for any polynomial g(d), gx-xg = -g'. (This is another way algebraists do calculus!) <br />
<br />
6c. Use the facts from (b) to show that the center of the Weyl algebra is just the field of constants. (Remark: you will use in the course of this proof the fact that a polynomial whose derivative is 0 is constant -- this would not be true if we were working in positive characteristic, as the example of x^p considered as a polynomial over F_p demonstrates.)<br />
<br />
<br />
7. If M is a left module for a ring R, we denote by Ann(M) the set of elements r in R such that rm = 0 for all m in M. Show that Ann(M) is a two-sided ideal. <br />
<br />
<br />
8. Let M be a left module for R, and let m be an element of M. We denote by Ann(m) the set of elements r in R such that rm = 0.<br />
<br />
8a. Show that Ann(m) is a left ideal, and show that the submodule Rm of M generated by m is isomorphic to the left module R/Ann(m).<br />
<br />
8c. Let V,W be as in question 6. Show that Ann(e^x) is the left ideal W(d-1).<br />
<br />
==HOMEWORK 11 (due Dec 13)==<br />
<br />
1. Let f: R -> S be a homomorphism of rings. As discussed in class, if M is a left R-module, then S tensor_R M is a left S-module, and if N is an S-module, then N can also be considered as an R-module by pullback of the action.<br />
<br />
Prove that if M is a finitely generated left R-module, then S tensor_R M is a finitely generated left S-module. On the other hand, give an example to show that N can be finitely generated as a left S-module but not finitely generated when considered as a left R-module.<br />
<br />
2. Recall from class that when V and W are complex vector spaces, W^dual tensor V is naturally isomorphic to the space of linear homomorphisms Hom(W,V).<br />
<br />
Now suppose that V and W are left C[G]-modules, i.e. complex representations of a group G. Show that W^dual tensor_C[G} V is the maximal G-invariant quotient of Hom(W,V); in other words, the quotient of Hom(W,V) by the subspace generated by all functions F of the form F(w) = f(gw) - gf(w) for some function f in Hom(W,V) and some g in G.<br />
<br />
3. '''How algebraists do logic.''' Why are Boolean rings called Boolean rings? Because they model some features of the Boolean arithmetic that governs propositional logic. Let A be a collection of propositions. We define two operations on A as follows. By a + b we mean the negation of a xor b, while by ab we mean "a or b."<br />
<br />
3a. It is not a priori obvious that these operations satisfy the ring axioms, but assume this for the moment. Show that a^2 = a for all a (so A is a boolean ring) and 2a = 0 for all a (so A has characteristic 2.)<br />
<br />
3b. Suppose we have a consistent assignment of truth-values to the propositions in A. Show that the set of true statements of A forms an ideal of A.<br />
<br />
3c. (Optional for people who like propositional logic) Show that a(b+c) = ab + ac with the given operations.<br />
<br />
<br />
4. Let K be Q[x]/(x^3-2).<br />
<br />
4a. Show that K is a field.<br />
<br />
4b. Show that K tensor_Q K decomposes as the direct sum of two fields. What are they? [Note: K tensor_Q K is a ring!]<br />
<br />
5. There was some discussion in class about when we can "go back and forth" between right modules and left modules, as is the case with group rings.<br />
<br />
5a. If R is a ring, we denote R^op ("the opposite ring") to be the ring whose elements are those of R, but whose multiplication law x_opp is defined by s x_opp r = rs. Show that if M is a left R-module, then the map M x R^op -> M sending (x,r) to rx makes M into a right R^op module. In particular, if there is a RING ISOMORPHISM from R to R^op, then every left R-module can be thought of as a right R-module.<br />
<br />
5b. Verify that the map sending g to g^{-1} extends by linearity to give a ring isomorphism between C[G] and C[G]^op.<br />
<br />
5c. Construct a ring isomorphism between M_n(Q) and M_n(Q)^op.<br />
<br />
5d. Optional: Give an example of a ring (with 1) which is not isomorphic to its opposite ring. (This is not so easy, I think! I found a MathOverflow thread with several examples, none of which came with a short proof of the non-isomorphism -- try your hand!)<br />
<br />
6. . Let R be a commutative ring with 1. If R has a unique maximal ideal M, R is called a "local ring".<br />
<br />
6a. If R is a local ring, show that every element of R that is not in M is a unit.<br />
<br />
6b. Conversely, if R is such that the set of nonunits forms an ideal I, show that R is a local ring with maximal ideal I.<br />
<br />
7. The inclusion of S_3 into S_4 associated to the inclusion of {1,2,3} into {1,2,3,4} induces a ring homomorphism from C[S_3} to C{S_4]. Let V be a representation of C{S_3]. Then, as discussed in class, V tensor_C[S_3] C[S_4] is a representation of S_4, called the '''induced representation''' of V.<br />
<br />
7a. Show that dim_C (V tensor_C[S_3] C[S_4]) = 4 dim_C V.<br />
<br />
7b. When V is the trivial representation of S_3, decompose (V tensor_C[S_3] C[S_4]) as a sum of irreducible representations of S_4.<br />
<br />
<!--==HOMEWORK PROBLEMS TO GIVE IN FUTURE SEMESTERS==<br />
<br />
Let f: C[x] tensor_C C[y] -> C[z] be the map of rings with f(x) = f(y) = z. Show that f(m tensor n) is not 0 for any pure tensor m tensor n, but f is not injective. (This is a warning not to try to check injectivity by checking that no pure tensors are killed by the map.)<br />
<br />
--></div>Dummithttps://hilbert.math.wisc.edu/wiki/index.php?title=741&diff=48127412012-12-22T10:33:25Z<p>Dummit: /* HOMEWORK 10 (due Dec 4) */</p>
<hr />
<div>'''Math 741'''<br />
<br />
Algebra<br />
<br />
Prof: [http://www.math.wisc.edu/~ellenber Jordan Ellenberg]<br />
<br />
Grader: Evan Dummit<br />
<br />
Ellenberg's office hours: Friday 3pm.<br />
<br />
Grader's office hours: Monday 4pm. Late homework may be given directly to the grader, along with either (i) the instructor's permission, or (ii) a polite request for mercy.<br />
<br />
This course, the first semester of the introductory graduate sequence in algebra, will cover the basic theory of groups, group actions, representation, linear and multilinear algebra, and the beginnings of ring theory.<br />
<br />
==SYLLABUS==<br />
<br />
In this space we will record the theorems and definitions we covered each week, which we can use as a list of notions you should be prepared to answer questions about on the Algebra qualifying exam. The material covered on the homework is also an excellent guide to the scope of the course. <br />
<br />
'''WEEK 1''': <br />
<br />
Definition of group. Associativity. Inverse. <br />
<br />
Examples of group: GL_n(R). GL_n(Z). Z/nZ. R. Z. R^*. The free group F_k on k generators. <br />
<br />
Homomorphisms. The homomorphisms from F_k to G are in bijection with G^k. Isomorphisms.<br />
<br />
'''WEEK 2''':<br />
<br />
The symmetric group (or permutation group) S_n on n letters. Cycle decomposition of a permutation. Order of a permutation. Thm: every element of a finite group has finite order. <br />
<br />
Subgroups. Left and right cosets. Lagrange's Theorem. Cyclic groups. The order of an element of a finite group is a divisor of the order of the group.<br />
<br />
The sign homomorphism S_n -> +-1.<br />
<br />
<br />
'''WEEK 3'''<br />
<br />
Normal subgroups. The quotient of a group by a normal subgroup. The first isomorphism theorem. Examples of S_n -> +-1 and S_4 -> S_3 with kernel V_4, the Klein 4-group.<br />
<br />
Centralizers and centers. Abelian groups. The center of SL_n(R) is either 1 or +-1.<br />
<br />
Groups with presentations. The infinite dihedral group <x,y | x^2 = 1, y^2 = 1>.<br />
<br />
'''WEEK 4'''<br />
<br />
More on groups with presentations.<br />
<br />
Second and third isomorphism theorems.<br />
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Semidirect products.<br />
<br />
'''WEEK 5'''<br />
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Group actions, orbits, and stabilizers.<br />
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Orbit-stabilizer theorem.<br />
<br />
Cayley's theorem.<br />
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Cauchy's theorem.<br />
<br />
'''WEEK 6'''<br />
<br />
Applications of orbit-stabilizer theorem (p-groups have nontrivial center, first Sylow theorem.)<br />
<br />
Classification of finite abelian groups and finitely generated abelian groups.<br />
<br />
Composition series and the Jordan-Holder theorem (which we state but don't prove.) <br />
<br />
The difference between knowing the composition factors and knowing the group (e.g. all p-groups of the same order have the same composition factors.)<br />
<br />
'''WEEK 7'''<br />
<br />
Simplicity of A_n.<br />
<br />
Nilpotent groups (main example: the Heisenberg group)<br />
<br />
Derived series and lower central series.<br />
<br />
Category theory 101: Definition of category and functor. Some examples. A group is a groupoid with one object.<br />
<br />
'''WEEK 8'''<br />
<br />
Introduction to representation theory.<br />
<br />
'''WEEK 10'''<br />
<br />
Ring theory 101: Rings, ring homomorphisms, ideals, isomorphism theorems. Examples: fields, Z, the Hamilton quaternions, matrix rings, rings of polynomials and formal power series, quadratic integer rings, group rings. Integral domains. Maximal and prime ideals. The nilradical.<br />
<br />
Module theory 101: Modules, module homomorphisms, submodules, isomorphism theorems. Noetherian modules and Zorn's lemma. Direct sums and direct products of arbitrary collections of modules.<br />
<br />
<br />
Below you will find a repository of homework problems. Note that some of these problems are taken from Lang's ''Algebra''.<br />
<br />
==HOMEWORK 1 (due Sep 20)==<br />
<br />
1. Suppose that H_1 and H_2 are subgroups of a group G. Prove that the intersection of H_1 and H_2 is a subgroup of G.<br />
<br />
2. Recall that S_3 (the symmetric group) is the group of permutations of the set {1..3}. List all the subgroups of S_3.<br />
<br />
3. We can define an equivalence relation on rational numbers by declaring two rational numbers to be equal whenever they differ by an integer. We denote the set of equivalence classes by Q/Z. The operation of addition makes Q/Z into a group.<br />
<br />
a) For each n, prove that Q/Z has a subgroup of order n.<br />
<br />
b) Prove that Q/Z is a ''divisible'' group: that is, if x is an element of Q/Z and n is an integer, there exists an element y of Q/Z such that ny = x. (Note that we write the operation in this group as addition rather than multiplication, which is why we write ny for the n-fold product of y with itself rather than y^n)<br />
<br />
c) Prove that Q/Z is not finitely generated. (Hint: prove that if x_1, .. x_d is a finite subset of Q/Z, the subgroup of Q/Z generated by x_1, ... x_d is finite.)<br />
<br />
4. Let F_2 be the free group on two generators x,y. Prove that there is a automorphism of F_2 which sends x to xyx and y to xy, and prove that this automorphism is unique.<br />
<br />
5. Let H be a subgroup of G, and let N_G(H), the ''normalizer'' of H in G, be the set of elements g in G satsifying g H g^{-1} = H. Prove that N_G(H) is a subgroup of G.<br />
<br />
6. Let T be the subgroup of GL_2(R) consisting of diagonal matrices. This is an example of a "Cartan subgroup," or "torus" (whence the notation T.) Describe the normalizer N(T) of T in GL_2(R). Prove that there is a homomorphism from N(T) to Z/2Z whose kernel is precisely T. <br />
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7. Prove that the group of inner automorphisms of a group is normal in the full automorphism group Aut(G).<br />
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8. Let H and H' be subgroups of a group G, and let x be an element of G. We denote by HxH' the set of elements of the form hxh', with h in H and h' in H', and we call HxH a ''double coset'' of the pair (H,H').<br />
<br />
a) Show that G decomposes as a disjoint union of double cosets of (H,H').<br />
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b) Let G be GL_2(R) and let B be the subgroup of upper triangular matrices. Prove that G decomposes into exactly two double cosets of (B,B).<br />
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c) Let G be the symmetric group S_n, and let H be the subgroup of permutations which fix 1. Describe the double coset decomposition of S_n into double cosets of (H,H).<br />
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==HOMEWORK 2 (due Sep 27)==<br />
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1. Prove that the symmetric group S_n is generated by transpositions. Hint: proceed by induction on n. Let H be the subgroup of S_n consisting of permutations fixing 1. Observe that H is isomorphic to S_{n-1}, so that H is generated by transpositions. Explain why it suffices to show that, for any g in S_n, the coset gH contains a transposition. Then explain why this is the case.<br />
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2. Given that the symmetric group S_n is generated by transpositions, show that there are no nontrivial homomorphisms from S_n to Z/pZ for any odd prime p.<br />
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3. Compute the centralizer of the triple flip (12)(34)(56) in S_6. (Note: this is the kind of problem that a computer algebra package like sage is very good at, and I encourage you to learn to use sage well enough to check your answer.)<br />
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4. Warning: a normal subgroup of a normal subgroup need not be normal! Let H be the subgroup of S_4 generated by (12)(34). Then H is a subgroup of the Klein 4-group V_4, which we have already shown is normal in S_4. Show that H is normal in V_4, but H is not normal in S_4.<br />
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5. The '''quaternion group''' Q is a non-abelian finite group of order 8. Its elements are 1, -1, i, -i, j, -j, k, -k; it satisfies g^2 = -1 for all g except +-1, and ijk = 1. (Note that the notation "-i" means "the product of i with -1" and -1 is understood to be central in Q.) Prove that every subgroup of Q is normal.<br />
<br />
6. (finishing example done in class) Let Gamma be the group F<x,y> / (x^2 = 1, y^2 = 1). (Remember, this means the quotient of the free group by the group generated by all conjugates of x^2 and y^2.) Prove that Gamma is infinite. Prove that Gamma has a subgroup R generated by xy isomorphic to Z, that R is normal, and that the quotient Gamma/R is isomorphic to Z/2Z. <br />
<br />
7. Suppose a group G has the property that every element in g satisfies g^2 = 1. Show that G must be abelian.<br />
<br />
==HOMEWORK 3 (due Oct 4)==<br />
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1. Multiplication by -1 is an automorphism of Z/nZ. Let a be the homomorphism Z/2Z -> Aut(Z/nZ) which sends the nontrivial element of Z/2Z to multiplication by -1. Then we have a semidirect product of Z/pZ by Z/2Z as defined in class. This group of order 2n is called a ''dihedral group.'' and is denoted D_n (or sometimes D_{2n}).<br />
<br />
1a. Compute the center of D_n. (Note that the answer depends on n!)<br />
1b. Show that all involutions of D_n are conjugate to each other if and only if n is odd.<br />
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2. (More on direct products that we didn't do in class.) Suppose that G is a semidirect product of N with H. Suppose furthermore that H is also normal in G. Prove that G is a direct product of N with H.<br />
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3. Let Q be the quaternion group from last week's homework. Show that Q does not decompose as a semidirect product N \rtimes H, apart form the trivial cases N = {e} and N = Q.<br />
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4. The ''affine linear group'' of degree n is the group of transformations from R^n to R^n of the form x -> Ax + b, where A is a matrix in GL_n(R) and b is a vector in R^n. Prove that the affine linear group is a semidirect product R^n \rtimes GL_n.<br />
<br />
5. The ''ordinary triangle group'' T(p,q,r) is the group with presentation <x,y | x^p = y^q = (xy)^r = 1>. This is a very interesting family of groups which arises naturally in many contexts in number theory and topology.<br />
<br />
6a. Prove that T(2,2,n) is isomorphic to the dihedral group D_n.<br />
6b. Prove that T(2,3,3) is finite and compute its order. Is it isomorphic to a finite group we've discussed in class?<br />
<br />
(It is a beautiful fact, but outside the scope of this course, that T(p,q,r) is finite if and only if 1/p + 1/q + 1/r > 1.)<br />
<br />
7. Let G be a subgroup of GL_n(R) which contains SL_n(R). Prove that G is a normal subgroup of GL_n(R).<br />
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8. Let G be a group and let H_1 and H_2 be subgroups of G of finite index (that is, the number of cosets in G/H_i is finite, i = 1,2.) Prove that H_1 intersect H_2 also has finite index in G.<br />
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9. Recall that the commutator [x,y] is x y x^{-1} y^{-1}. If G is a group, the commutator subgroup of G, denoted G' or [G,G], is the subgroup of G generated by all commutators [x,y] for x,y in G.<br />
<br />
9a. Show that G' is a normal subgroup of G.<br />
9b. Show that G/G' is an abelian group.<br />
9c. Show that if f: G -> A is a homomorphism, with A abelian, then G' is contained in ker f. Conclude that f factors through a homomorphism G/G' -> A.<br />
9d. Show that G' = G if and only if G has no nontrivial abelian quotient. In this case, we say G is ''perfect''.<br />
<br />
10. Under what conditions on (p,q,r) is the triangle group T(p,q,r) perfect?<br />
<br />
==HOMEWORK 4 (due Oct 16)==<br />
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1. Let X be a set with a transitive action of a group G, which we think of as a homomorphism f: G -> Sym(X). Let H_x be the stabilizer of an element x of X.<br />
<br />
1a. If x' is another element of X, show that H_{x"} and H_x are conjugate subgroups. (Hint: this is not true without the hypothesis of transitivity, so make sure to use it!)<br />
<br />
1b. Show that the kernel of f is the intersection of all the conjugates of H_x in G. (This subgroup is called the ''normal core'' of H_x.)<br />
<br />
2. This week, a certified mathematical grown-up asked on MathOverflow what the index-3 subgroups of SL_2(Z/pZ) were. Let's do it ourselves for homework. Let p be a prime, and suppose H is an index-3 subgroup of SL_2(Z/pZ). I do not claim that the proof sketched below is necessarily the easiest, but it allows us to cover some ground that we won't get to do in lecture.<br />
<br />
2a. Using the coset action, show that the normal core of H is the kernel of a homomorphism f from SL_2(Z/pZ) to S_3.<br />
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2b. A ''unipotent matrix'' is a matrix A such that (A-I)^n = 0 for some positive integer n. Prove that a unipotent element of SL_2(Z/pZ) has order dividing p.<br />
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2c. You may use the fact that unipotent matrices generate SL_2(Z/pZ). Given this fact, show that f must be trivial when p > 3.<br />
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2d. Using the above, show that SL_2(Z/pZ) has no index-3 subgroups when p > 3.<br />
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2e. (extra credit) What can you say about the minimal index of a proper subgroup of SL_2(Z/pZ)?<br />
<br />
3. Let H be a subgroup of G of index 2. Prove that H is normal.<br />
<br />
4. We proved two theorems in class: Lagrange's theorem, which says that the order of a subgroup of G divides |G|, and Cauchy's theorem, which says that if p divides |G| then there exists a subgroup of G of order p. Cauchy's theorem is a kind of partial converse to Lagrange's theorem, but the full converse is false. Give an example of a finite group G and a divisor n of |G| such that you can prove there is no subgroup of G of order n.<br />
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5. Let X be the set of lines in F_3^2 (here a line means a 1-dimensional linear subspace) so that |X| = 4. Let G be the projective linear group PGL_2(F_3), so that G acts on X via its action on F_3^2. This action can be thought of as a homomorphism from G to S_4. Show that this homomorphism is an ISOMORPHISM from PGL_2(F_3) to S_4.<br />
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6. The group S_3 has only three conjugacy classes. Prove that a finite group with at most three conjugacy classes has order at most 6.<br />
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7. Let X be the set of ordered triples of elements of {1,..,n}, for some n >= 3. Then S_n acts on X. How many orbits are there? (Hint: the answer does not depend on n.)<br />
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8. Suppose that G and H are groups and f: G -> H is a homomorphism. Recall the definition of the abelianization G^ab from the previous problem set.<br />
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8a. Show that the composition G -> H -> H^ab factors through a unique homomorphism G^ab -> H^ab, which we denote f^ab.<br />
<br />
8b. Show that if f: G -> H and g: H -> Q are homomorphisms, then f^ab o g^ab = (f o g)^ab.<br />
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(For those reading MacLane, this constitutes a proof that abelianization is a '''functor''' from the category of groups to the category of abelian groups.)<br />
<br />
==HOMEWORK 5 (due Oct 23)==<br />
<br />
1. Show that there are only two different isomorphism classes of groups of order 14. (Hint: prove that such a group must be a semidirect product of a group of order 7 by a group of order 2.)<br />
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2. (variant of Burnside's Lemma) Let G be a finite group and H a normal subgroup. Suppose G acts on a finite set X. <br />
<br />
2a. Show that if x and y lie in the same H-orbit, and g is an element of G, then gx and gy also lie in the same H-orbit. Thus, G/H has a natural action on the set of H-orbits X/H.<br />
<br />
2b. Show that the average number of fixed points in X of an element of the coset gH is the same as the number of fixed points of the element gH of G/H in its action on X/H.<br />
<br />
3. A ''central extension'' of a group G by an abelian group A is a group E, together with a surjective homomorphism E -> G whose kernel is central in E and is isomorphic to A.<br />
<br />
3a. Show that the quaternion group is a central extension of (Z/2Z)^2 by Z/2Z.<br />
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3b. Give an example of another nonabelian group which is a central extension of (Z/2Z)^2 by Z/2Z and which is not isomorphic to the quaternion group.<br />
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4. Prove that a central extension of an abelian group is nilpotent.<br />
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5. Give two different composition series for S_4 and show that they have the same composition factors.<br />
<br />
6. Suppose that A is a finitely generated abelian group, and B is a subgroup of A.<br />
<br />
6a. Prove that B is finitely generated and that rank(B) <= rank(A).<br />
<br />
6b. Prove that rank(A/B) = rank(A) - rank(B).<br />
<br />
7. For this problem, you may use the fact (which I should have asked on homework earlier) that if H_1 and H_2 are subgroups of G, then<br />
<br />
[G: H_1 intersect H_2] <= [G:H_1][G:H_2].<br />
<br />
Prove that if G is a simple group, d is an integer greater than 1, and S is a subgroup of index d in G, then |G| <= d^d.<br />
<br />
Conclude that, in particular, A_5 has no subgroup of index 3. On the other hand, exhibit a subgroup of A_5 of index 5.<br />
<br />
(Note: d^d is not a sharp bound; can you get a bound of d factorial instead?)<br />
<br />
<!--(This argument shows that a solvable subgroup of A_n has size at most about (n-1)!, but this is far from sharp; e.g. I found a citation to a 1967 paper of Dixon which shows that a solvable subgroup of A_n has size at most k^n where k is around 2.88.)--><br />
<br />
8. Give an example of an abelian group A with a subgroup B such that B is not a direct summand of A.<br />
<br />
9. Let A be the subgroup of Z^2 generated by the vector (1,0). Let B be the subgroup of Z^2 generated by the vector (3,3). Show that A and B are isomorphic to each other, but Z^2 / A is not isomorphic to Z^2 / B.<br />
<br />
==HOMEWORK 6 (due Oct 30)==<br />
<br />
1. In this problem, you will construct a functor F from the category of finite sets to the category of complex vector spaces. I'll tell you what it does on objects: if X is a finite set, then F(X) is the vector space spanned by a set of basis elements {e_x}_{x in X}, so that dim F(X) = |X|. Your job: complete the definition of F by describing the map from Mor_{Finite Sets}(X,Y) to Mor_{Vector Spaces}(F(X),F(Y)), and showing that your map respects composition.<br />
<br />
2. If (V, rho) is a representation of a group G, denote by V^G the space of vectors in V such that gv = v for all g in G; these are called _invariant_ vectors. We saw that the permutation representation of S_3 has a one-dimensional space of invariants. Prove that if X is a finite set with G-action, and V_X the corresponding permutation representation of G, then<br />
<br />
dim V_X^G = number of orbits of X.<br />
<br />
3. (Invariants are functorial) Let G be a group. Then there is a category called C[G]-Mod whose objects are complex representations of G (i.e. pairs (V,rho) where V is a complex vector space and rho is a homomorphism from G to GL(V)). The morphisms in this category are the ones described in class: Mor((V,rho),(W,psi)) is the set of linear maps f from V to W such that<br />
<br />
f(rho(g)(v)) = psi(g)(f(v))<br />
<br />
for all g in G and all v in V.<br />
<br />
Show that there is a functor H_0 from the category of representations of G to the category of vector spaces whose action on objects is given by<br />
<br />
H_0((V) = V^G.<br />
<br />
(In other words: your job again is to explain how to associate to a morphism from (V,rho) to (W,phi) a morphism from V^G to W^G, in a way that's compatible with composition.)<br />
<br />
<!--4. The group PSL_2(Z) has a presentation of the form [U,V|U^2 = V^3 = 1]. Thus, an n-dimensional representation of PSL_2(Z) is precisely a pair of n x n matrices (rho(U),rho(V)), such that rho(U)^2 = rho(V)^3 = 1. Prove that for every n > 1 there are uncountably many isomorphism classes of n-dimensional representations of PSL_2(Z). (Extra credit: show that there are uncountably many isomorphism classes of _irreducible_ representations.)<br />
<br />
An amazing theorem of Margulis (superrigidity) shows that SL_2(Z) is very special among arithmetic groups in this way. For instance, there are only _finitely_ many isomorphism classes of n-dimensional representations of PSL_d(Z) for any d > 2; those groups are "rigid."--><br />
<br />
5. Let G be a finite group, let X be a set with G-action, and let chi be the character of the corresponding permutation representation. Show that, for each g in G, chi(g) is the number of elements of X fixed by g. (I am actually going to prove this in class.)<br />
<br />
6. Let V be the space of homogeneous degree-3 polynomials in variables x_1, x_2, x_3; then S_3 acts on V by permutation of the three variables. Describe the decomposition of V into irreducible representations of S_3.<br />
<br />
7a. Show that if chi is the character of a representation of a finite group G, then chi(g^{-1}) is the complex conjugate of chi.<br />
<br />
7b. As a consequence, show that if g is conjugate to g^{-1} for all g in G, then chi(g) is real.<br />
<br />
7c. The groups D_p and S_n have the property that g is conjugate to g^{-1} for _every_ g in G, so that every character is a real-valued function on G. Give an example of a group G where this is not the case, and give an example of a representation of G whose character takes non-real values.<br />
<br />
==HOMEWORK 7 (due Nov 6)==<br />
<br />
1. Let G be a finite group, let X be a finite set with G-action, and let chi be the character of the corresponding permutation representation. Show that, for each g in G, chi(g) is the number of elements of X fixed by g. (I am actually going to prove this in class.)<br />
<br />
2a. Show that if chi is the character of a finite-dimensional representation of a finite group G, then chi(g^{-1}) is the complex conjugate of chi(g) for all g.<br />
<br />
2b. As a consequence, show that if g is conjugate to g^{-1} for all g in G, then chi(g) is real.<br />
<br />
2c. The groups D_p and S_n have the property that g is conjugate to g^{-1} for _every_ g in G, so that every character is a real-valued function on G. Give an example of a group G where this is not the case, and give an example of a representation of G whose character takes non-real values.<br />
<br />
3. Let V and W be complex vector spaces. Let f be a linear transformation of V, and g a linear transformation of W. <br />
<br />
3a. Show that there is a unique linear transformation F satisfying<br />
<br />
F(v tensor w) = f(v) tensor g(w)<br />
<br />
for all v in V and all w in W. We denote this transformation by f tensor g.<br />
<br />
3b. Suppose V and W are finite dimensional. Show that Trace(f tensor g) = Trace(f) Trace(g). What does this say when f and g are both the identity transformation?<br />
<br />
4. Let V be a vector space. We define Sym^2 V to be the quotient of V tensor V by the subspace generated by all elements of the form<br />
<br />
v tensor w - w tensor v<br />
<br />
for v,w in V.<br />
<br />
Suppose dim V = n. What is dim Sym^2 V?<br />
<br />
5. Show that Z/pZ tensor Z/qZ is zero when p and q are distinct primes.<br />
<br />
==HOMEWORK 8 (due Nov 13)==<br />
<br />
1. (Schur's lemma can be false over non-algebraically closed fields) Let G be Z/3Z and let g be a generator of G. Then G acts by cyclic permutations on the set {1,2,3}; let P be the corresponding permutation representation on the REAL (not complex!) vector space generated by e_1,e_2,e_3.<br />
<br />
1a. Show that P breaks up as the direct sum of the trivial representation and a 2-dimensional representation V which is irreducible. (Remember, we are not working over the complex numbers, so you cannot use the criterion for irreducibility in terms of the character that we worked out in class for the complex case; you need to prove directly that V has no nontrivial proper subrepresentation.)<br />
<br />
1b. Show that the map from V to V induced by g is G-equivariant, but is neither zero nor a scalar multiplication.<br />
<br />
2. If V_1 and V_2 are representations (not necessarily irreducible!) of G, show that <chi_V_1, chi V_2> is a non-negative integer.<br />
<br />
3. When a formula always yields a non-negative integer, it is often because it is secretly a formula for either the cardinality of some finite set or the dimension of some finite-dimensional vector space. In this case, show that<br />
<br />
<chi_{V_1}, chi_{V_2}> = dim_C Hom_G(V_1,V_2)<br />
<br />
where we recall that Hom_G(V_1,G_2) refers to the space of G-equivariant homomorphisms from V_1 to V_2.<br />
<br />
4. If A is an abelian group, show that every irreducible representation of A is 1-dimensional. (Hint: show that there is a vector which is an eigenvector for every element of A!)<br />
<br />
5. Let G be the group of order 20 which is the semidirect product of A by H, where A is Z/5Z and H is (Z/5Z)^* with its natural action on A. We will work out all the irreducible representations of A. <br />
<br />
5a. First of all, consider the 5-dimensional permutation representation of G acting by left multiplication on the cosets of H. Show that this representation decomposes as a direct sum of the trivial representation and an irreducible 4-dimensional representation. (Hint: it is probably easier to use the character of this representation than to prove irreducibility directly.) <br />
<br />
5b. Now show that there are 4 1-dimensional representations of G which factor through the quotient G/A (i.e. such that rho(A) is the identity.)<br />
<br />
5c. Using the decomposition of the regular representation, show that the irreducible representations of G you have constructed are the only ones.<br />
<br />
==HOMEWORK 9 (due Nov 20)==<br />
<br />
0. Find all ring homomorphisms from Z to Z/30Z. [Warning: there are 8 of them, not 30.]<br />
<br />
<br />
1. If R is a ring (not necessarily with 1) such that a^2 = a for every a in R, R is called a Boolean ring.<br />
<br />
1a. Show that any nonzero Boolean ring has characteristic 2. (In other words, show that a+a=0 for all a in R.)<br />
<br />
1b. Show that every Boolean ring is commutative.<br />
<br />
<br />
2. Let phi : R -> S be a homomorphism of commutative rings with 1.<br />
<br />
2a. If I is an ideal of S, show that phi^(-1) (I) is an ideal of R.<br />
<br />
2b. If P is a prime ideal of S, show that phi^(-1) (P) is either R or a prime ideal of R.<br />
<br />
2c. If M is a maximal ideal of S and phi is surjective, show that phi^(-1) (M) is a maximal ideal of R.<br />
<br />
2d. Find a homomorphism phi : R -> S and a maximal ideal M of S such that phi^(-1) (M) is not a maximal ideal of R.<br />
<br />
<br />
3. Let Nil(A) denote the set of nilpotent elements of a ring A, and let R be a commutative ring. In class it was observed that Nil(R) is an ideal of R.<br />
<br />
3a. Find (a generator of) Nil(Z/720Z).<br />
<br />
3b. Show that Nil(R/Nil(R)) = 0.<br />
<br />
3c. If x is in Nil(R) and u is a unit, show that u+x is a unit.<br />
<br />
3d. Show that Nil(S), where S is the 2x2 matrices with real coefficients, is not an ideal of S.<br />
<br />
<br />
4a. If M is an R-module and I is an ideal of R such that im = 0 for all i in I and m in M, show that M carries the structure of an R/I-module.<br />
<br />
4b. If S is a commutative ring with 1 and J is a maximal ideal of S, show that J/J^2 is a vector space over S/J.<br />
<br />
<br />
5. A "divisible element" x of a Z-module is one such that for each positive integer k, there exists a y such that ky=x. Show that a free Z-module cannot contain a nonzero divisible element.<br />
<br />
<br />
6. Prove that the direct sum of any collection of free R-modules is also a free R-module.<br />
<br />
<br />
7. In this problem you will prove that the module M given by the direct product of copies of Z, indexed by the positive integers, is not a free Z-module. Therefore, assume that M is a free Z-module with basis B, and let N be the direct sum of copies of Z, as a submodule of M.<br />
<br />
7a. Show there is a countable subset B1 of B such that N is contained in the submodule N1 generated by B1. [Hint: N is countable.]<br />
<br />
7b. Show that M/N1 is a free Z-module.<br />
<br />
7c. Let S = {(b1, b2, b3, ...) : b_i = i! or -i!}. Show that there is some s in S which is not in N1. [Hint: S is uncountable.]<br />
<br />
7d. For any s in S but not N1, show that for each positive integer k, there exists some m in M with s-km in N1. Conclude that s+N1 is a divisible element of M/N1.<br />
<br />
7e. Use problem 5 to obtain a contradiction.<br />
<br />
<br />
8. In class I said "rank" is not always well-defined for R-modules if R is not commutative. Here is an example demonstrating this assertion. Let M be the countably infinite direct product of copies of Z as in exercise 7, and let R be the endomorphism ring of M. Define phi_1(a1,a2,a3,a4,...) = (a1,a3,a5,...) and phi_2(a1,a2,a3,a4,...) = (a2,a4,a6,...).<br />
<br />
8a. Show that {phi_1, phi_2} is a free basis for R, as a left R-module. [Hint: let psi_1(a1,a2,a3,...) = (a1,0,a2,0,...) and psi_2(a1,a2,a3,...) = (0,a1,0,a2,...). Compute the products of the phi_i and psi_j in either order, and use the result to show that the phi_i are a basis.]<br />
<br />
8b. Show that R is isomorphic to R^n for every positive integer n.<br />
<br />
<br />
==HOMEWORK 10 (due Dec 4)==<br />
<br />
1. '''How algebraists do calculus'''. The ''ring of dual numbers'' is the ring R = C[e]/(e^2). We think of e as an infinitesimal number; that is, something so small that its square is 0. This algebraic notion of infinitesimal is the basis for the algebraic notion of differentiation, which works as follows. Let f be a polynomial in R[x]. Show that there is a unique polynomial g in C[x] such that<br />
<br />
f(x+e) - f(x) = eg(x)<br />
<br />
and that in fact g is the derivative of f. In other words, the usual definition of "derivative" works just fine in this context, without any use of the notion of limit!<br />
<br />
2. Write down all the ideals of C[e]/e^2. (There are three.)<br />
<br />
3. Show that any right ideal of the ring M_2(Q) of 2x2 matrices is an ideal of the form I_V described in class for some subspace V of Q^2; that is, it consists precisely of those linear transformations whose image is contained in V.<br />
<br />
<br />
4. An ''idempotent'' element of a ring is an element e satisfying e^2 = e.<br />
<br />
4a. If e is an idempotent, show that 1-e is also an idempotent.<br />
<br />
4b. Suppose R has a unit (we are actually always supposing this, but I just want to remind you of it for this problem) and suppose e is a ''central'' idempotent, i.e. an idempotent which commutes with all elements of R. Show that eR and (1-e)R are both two-sided ideals. Moreover, show that eR and (1-e)R are also subrings of R (with the warning that the identity in eR is not 1, which is not contained in eR, but rather e.)<br />
<br />
4c. Show that the map f_e: R -> eR defined by f_e(r) = er is a ring homomorphism. Finally show that the ring homomorphism<br />
<br />
f_{e} x f_{1-e}: R -> eR x (1-e)R <br />
<br />
is actually an isomorphism of rings. In other words, the presence of a central idempotent induces a decomposition a ring as a direct product.<br />
<br />
<br />
5. The ''center'' of a ring R is the subring of elements commuting with every element of R.<br />
<br />
5a. Show that the center of M_n(Q) is the ring of scalar matrices.<br />
<br />
5b. Show that the center of the group ring C[G] is the set of elements sum f(g) g such that f(g) is a class function.<br />
<br />
<br />
6. '''How algebraists do differential equations.''' We define the ''Weyl algebra'' W of differential operators in one variable as follows. Multiplication by x is a linear transformation on the complex vector space V of all smooth functions on C. Differentiation, which we denote d, is also a linear transformation on V. So we let W be the subring of End(V) generated by x and d. We note that x and d satisfy the relation xd = dx - 1; in particular, the Weyl algebra is not commutative.<br />
<br />
6a. Describe the submodule of V which is annihilated by the element d-x in W.<br />
<br />
6b. Show that, for any polynomial f(x), fd-df = -f', and also show that for any polynomial g(d), gx-xg = -g'. (This is another way algebraists do calculus!) <br />
<br />
6c. Use the facts from (b) to show that the center of the Weyl algebra is just the field of constants. (Remark: you will use in the course of this proof the fact that a polynomial whose derivative is 0 is constant -- this would not be true if we were working in positive characteristic, as the example of x^p considered as a polynomial over F_p demonstrates.)<br />
<br />
<br />
7. If M is a left module for a ring R, we denote by Ann(M) the set of elements r in R such that rm = 0 for all m in M. Show that Ann(M) is a two-sided ideal. <br />
<br />
<br />
8. Let M be a left module for R, and let m be an element of M. We denote by Ann(m) the set of elements r in R such that rm = 0.<br />
<br />
8a. Show that Ann(m) is a left ideal, and show that the submodule Rm of M generated by m is isomorphic to the left module R/Ann(m).<br />
<br />
8c. Let V,W be as in question 6. Show that Ann(e^x) is the left ideal W(d-1).<br />
<br />
==HOMEWORK 11 (due Dec 13)==<br />
<br />
1. Let f: R -> S be a homomorphism of rings. As discussed in class, if M is a left R-module, then S tensor_R M is a left S-module, and if N is an S-module, then N can also be considered as an R-module by pullback of the action.<br />
<br />
Prove that if M is a finitely generated left R-module, then S tensor_R M is a finitely generated left S-module. On the other hand, give an example to show that N can be finitely generated as a left S-module but not finitely generated when considered as a left R-module.<br />
<br />
2. Recall from class that when V and W are complex vector spaces, W^dual tensor V is naturally isomorphic to the space of linear homomorphisms Hom(W,V).<br />
<br />
Now suppose that V and W are left C[G]-modules, i.e. complex representations of a group G. Show that W^dual tensor_C[G} V is the maximal G-invariant quotient of Hom(W,V); in other words, the quotient of Hom(W,V) by the subspace generated by all functions F of the form F(w) = f(gw) - gf(w) for some function f in Hom(W,V) and some g in G.<br />
<br />
3. '''How algebraists do logic.''' Why are Boolean rings called Boolean rings? Because they model some features of the Boolean arithmetic that governs propositional logic. Let A be a collection of propositions. We define two operations on A as follows. By a + b we mean the negation of a xor b, while by ab we mean "a or b."<br />
<br />
3a. It is not a priori obvious that these operations satisfy the ring axioms, but assume this for the moment. Show that a^2 = a for all a (so A is a boolean ring) and 2a = 0 for all a (so A has characteristic 2.)<br />
<br />
3b. Suppose we have a consistent assignment of truth-values to the propositions in A. Show that the set of true statements of A forms an ideal of A.<br />
<br />
3c. (Optional for people who like propositional logic) Show that a(b+c) = ab + ac with the given operations.<br />
<br />
<br />
4. Let K be Q[x]/(x^3-2).<br />
<br />
4a. Show that K is a field.<br />
<br />
4b. Show that K tensor_Q K decomposes as the direct sum of two fields. What are they?<br />
<br />
5. There was some discussion in class about when we can "go back and forth" between right modules and left modules, as is the case with group rings.<br />
<br />
5a. If R is a ring, we denote R^op ("the opposite ring") to be the ring whose elements are those of R, but whose multiplication law x_opp is defined by s x_opp r = rs. Show that if M is a left R-module, then the map M x R^op -> M sending (x,r) to rx makes M into a right R^op module. In particular, if there is a RING ISOMORPHISM from R to R^op, then every left R-module can be thought of as a right R-module.<br />
<br />
5b. Verify that the map sending g to g^{-1} extends by linearity to give a ring isomorphism between C[G] and C[G]^op.<br />
<br />
5c. Construct a ring isomorphism between M_n(Q) and M_n(Q)^op.<br />
<br />
5d. Optional: Give an example of a ring which is not isomorphic to its opposite ring. (This is not so easy, I think! I found a MathOverflow thread with several examples, none of which came with a short proof of the non-isomorphism -- try your hand!)<br />
<br />
6. . Let R be a commutative ring with 1. If R has a unique maximal ideal M, R is called a "local ring".<br />
<br />
6a. If R is a local ring, show that every element of R that is not in M is a unit.<br />
<br />
6b. Conversely, if R is such that the set of nonunits forms an ideal I, show that R is a local ring with maximal ideal I.<br />
<br />
7. The inclusion of S_3 into S_4 associated to the inclusion of {1,2,3} into {1,2,3,4} induces a ring homomorphism from C[S_3} to C{S_4]. Let V be a representation of C{S_3]. Then, as discussed in class, V tensor_C[S_3] C[S_4] is a representation of S_4, called the '''induced representation''' of V.<br />
<br />
7a. Show that dim_C (V tensor_C[S_3] C[S_4]) = 4 dim_C V.<br />
<br />
7b. When V is the trivial representation of S_3, decompose (V tensor_C[S_3] C[S_4]) as a sum of irreducible representations of S_4.<br />
<br />
<!--==HOMEWORK PROBLEMS TO GIVE IN FUTURE SEMESTERS==<br />
<br />
Let f: C[x] tensor_C C[y] -> C[z] be the map of rings with f(x) = f(y) = z. Show that f(m tensor n) is not 0 for any pure tensor m tensor n, but f is not injective. (This is a warning not to try to check injectivity by checking that no pure tensors are killed by the map.)<br />
<br />
--></div>Dummithttps://hilbert.math.wisc.edu/wiki/index.php?title=741&diff=48117412012-12-22T10:30:34Z<p>Dummit: /* HOMEWORK 5 (due Oct 23) */</p>
<hr />
<div>'''Math 741'''<br />
<br />
Algebra<br />
<br />
Prof: [http://www.math.wisc.edu/~ellenber Jordan Ellenberg]<br />
<br />
Grader: Evan Dummit<br />
<br />
Ellenberg's office hours: Friday 3pm.<br />
<br />
Grader's office hours: Monday 4pm. Late homework may be given directly to the grader, along with either (i) the instructor's permission, or (ii) a polite request for mercy.<br />
<br />
This course, the first semester of the introductory graduate sequence in algebra, will cover the basic theory of groups, group actions, representation, linear and multilinear algebra, and the beginnings of ring theory.<br />
<br />
==SYLLABUS==<br />
<br />
In this space we will record the theorems and definitions we covered each week, which we can use as a list of notions you should be prepared to answer questions about on the Algebra qualifying exam. The material covered on the homework is also an excellent guide to the scope of the course. <br />
<br />
'''WEEK 1''': <br />
<br />
Definition of group. Associativity. Inverse. <br />
<br />
Examples of group: GL_n(R). GL_n(Z). Z/nZ. R. Z. R^*. The free group F_k on k generators. <br />
<br />
Homomorphisms. The homomorphisms from F_k to G are in bijection with G^k. Isomorphisms.<br />
<br />
'''WEEK 2''':<br />
<br />
The symmetric group (or permutation group) S_n on n letters. Cycle decomposition of a permutation. Order of a permutation. Thm: every element of a finite group has finite order. <br />
<br />
Subgroups. Left and right cosets. Lagrange's Theorem. Cyclic groups. The order of an element of a finite group is a divisor of the order of the group.<br />
<br />
The sign homomorphism S_n -> +-1.<br />
<br />
<br />
'''WEEK 3'''<br />
<br />
Normal subgroups. The quotient of a group by a normal subgroup. The first isomorphism theorem. Examples of S_n -> +-1 and S_4 -> S_3 with kernel V_4, the Klein 4-group.<br />
<br />
Centralizers and centers. Abelian groups. The center of SL_n(R) is either 1 or +-1.<br />
<br />
Groups with presentations. The infinite dihedral group <x,y | x^2 = 1, y^2 = 1>.<br />
<br />
'''WEEK 4'''<br />
<br />
More on groups with presentations.<br />
<br />
Second and third isomorphism theorems.<br />
<br />
Semidirect products.<br />
<br />
'''WEEK 5'''<br />
<br />
Group actions, orbits, and stabilizers.<br />
<br />
Orbit-stabilizer theorem.<br />
<br />
Cayley's theorem.<br />
<br />
Cauchy's theorem.<br />
<br />
'''WEEK 6'''<br />
<br />
Applications of orbit-stabilizer theorem (p-groups have nontrivial center, first Sylow theorem.)<br />
<br />
Classification of finite abelian groups and finitely generated abelian groups.<br />
<br />
Composition series and the Jordan-Holder theorem (which we state but don't prove.) <br />
<br />
The difference between knowing the composition factors and knowing the group (e.g. all p-groups of the same order have the same composition factors.)<br />
<br />
'''WEEK 7'''<br />
<br />
Simplicity of A_n.<br />
<br />
Nilpotent groups (main example: the Heisenberg group)<br />
<br />
Derived series and lower central series.<br />
<br />
Category theory 101: Definition of category and functor. Some examples. A group is a groupoid with one object.<br />
<br />
'''WEEK 8'''<br />
<br />
Introduction to representation theory.<br />
<br />
'''WEEK 10'''<br />
<br />
Ring theory 101: Rings, ring homomorphisms, ideals, isomorphism theorems. Examples: fields, Z, the Hamilton quaternions, matrix rings, rings of polynomials and formal power series, quadratic integer rings, group rings. Integral domains. Maximal and prime ideals. The nilradical.<br />
<br />
Module theory 101: Modules, module homomorphisms, submodules, isomorphism theorems. Noetherian modules and Zorn's lemma. Direct sums and direct products of arbitrary collections of modules.<br />
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Below you will find a repository of homework problems. Note that some of these problems are taken from Lang's ''Algebra''.<br />
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==HOMEWORK 1 (due Sep 20)==<br />
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1. Suppose that H_1 and H_2 are subgroups of a group G. Prove that the intersection of H_1 and H_2 is a subgroup of G.<br />
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2. Recall that S_3 (the symmetric group) is the group of permutations of the set {1..3}. List all the subgroups of S_3.<br />
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3. We can define an equivalence relation on rational numbers by declaring two rational numbers to be equal whenever they differ by an integer. We denote the set of equivalence classes by Q/Z. The operation of addition makes Q/Z into a group.<br />
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a) For each n, prove that Q/Z has a subgroup of order n.<br />
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b) Prove that Q/Z is a ''divisible'' group: that is, if x is an element of Q/Z and n is an integer, there exists an element y of Q/Z such that ny = x. (Note that we write the operation in this group as addition rather than multiplication, which is why we write ny for the n-fold product of y with itself rather than y^n)<br />
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c) Prove that Q/Z is not finitely generated. (Hint: prove that if x_1, .. x_d is a finite subset of Q/Z, the subgroup of Q/Z generated by x_1, ... x_d is finite.)<br />
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4. Let F_2 be the free group on two generators x,y. Prove that there is a automorphism of F_2 which sends x to xyx and y to xy, and prove that this automorphism is unique.<br />
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5. Let H be a subgroup of G, and let N_G(H), the ''normalizer'' of H in G, be the set of elements g in G satsifying g H g^{-1} = H. Prove that N_G(H) is a subgroup of G.<br />
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6. Let T be the subgroup of GL_2(R) consisting of diagonal matrices. This is an example of a "Cartan subgroup," or "torus" (whence the notation T.) Describe the normalizer N(T) of T in GL_2(R). Prove that there is a homomorphism from N(T) to Z/2Z whose kernel is precisely T. <br />
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7. Prove that the group of inner automorphisms of a group is normal in the full automorphism group Aut(G).<br />
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8. Let H and H' be subgroups of a group G, and let x be an element of G. We denote by HxH' the set of elements of the form hxh', with h in H and h' in H', and we call HxH a ''double coset'' of the pair (H,H').<br />
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a) Show that G decomposes as a disjoint union of double cosets of (H,H').<br />
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b) Let G be GL_2(R) and let B be the subgroup of upper triangular matrices. Prove that G decomposes into exactly two double cosets of (B,B).<br />
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c) Let G be the symmetric group S_n, and let H be the subgroup of permutations which fix 1. Describe the double coset decomposition of S_n into double cosets of (H,H).<br />
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==HOMEWORK 2 (due Sep 27)==<br />
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1. Prove that the symmetric group S_n is generated by transpositions. Hint: proceed by induction on n. Let H be the subgroup of S_n consisting of permutations fixing 1. Observe that H is isomorphic to S_{n-1}, so that H is generated by transpositions. Explain why it suffices to show that, for any g in S_n, the coset gH contains a transposition. Then explain why this is the case.<br />
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2. Given that the symmetric group S_n is generated by transpositions, show that there are no nontrivial homomorphisms from S_n to Z/pZ for any odd prime p.<br />
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3. Compute the centralizer of the triple flip (12)(34)(56) in S_6. (Note: this is the kind of problem that a computer algebra package like sage is very good at, and I encourage you to learn to use sage well enough to check your answer.)<br />
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4. Warning: a normal subgroup of a normal subgroup need not be normal! Let H be the subgroup of S_4 generated by (12)(34). Then H is a subgroup of the Klein 4-group V_4, which we have already shown is normal in S_4. Show that H is normal in V_4, but H is not normal in S_4.<br />
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5. The '''quaternion group''' Q is a non-abelian finite group of order 8. Its elements are 1, -1, i, -i, j, -j, k, -k; it satisfies g^2 = -1 for all g except +-1, and ijk = 1. (Note that the notation "-i" means "the product of i with -1" and -1 is understood to be central in Q.) Prove that every subgroup of Q is normal.<br />
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6. (finishing example done in class) Let Gamma be the group F<x,y> / (x^2 = 1, y^2 = 1). (Remember, this means the quotient of the free group by the group generated by all conjugates of x^2 and y^2.) Prove that Gamma is infinite. Prove that Gamma has a subgroup R generated by xy isomorphic to Z, that R is normal, and that the quotient Gamma/R is isomorphic to Z/2Z. <br />
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7. Suppose a group G has the property that every element in g satisfies g^2 = 1. Show that G must be abelian.<br />
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==HOMEWORK 3 (due Oct 4)==<br />
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1. Multiplication by -1 is an automorphism of Z/nZ. Let a be the homomorphism Z/2Z -> Aut(Z/nZ) which sends the nontrivial element of Z/2Z to multiplication by -1. Then we have a semidirect product of Z/pZ by Z/2Z as defined in class. This group of order 2n is called a ''dihedral group.'' and is denoted D_n (or sometimes D_{2n}).<br />
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1a. Compute the center of D_n. (Note that the answer depends on n!)<br />
1b. Show that all involutions of D_n are conjugate to each other if and only if n is odd.<br />
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2. (More on direct products that we didn't do in class.) Suppose that G is a semidirect product of N with H. Suppose furthermore that H is also normal in G. Prove that G is a direct product of N with H.<br />
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3. Let Q be the quaternion group from last week's homework. Show that Q does not decompose as a semidirect product N \rtimes H, apart form the trivial cases N = {e} and N = Q.<br />
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4. The ''affine linear group'' of degree n is the group of transformations from R^n to R^n of the form x -> Ax + b, where A is a matrix in GL_n(R) and b is a vector in R^n. Prove that the affine linear group is a semidirect product R^n \rtimes GL_n.<br />
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5. The ''ordinary triangle group'' T(p,q,r) is the group with presentation <x,y | x^p = y^q = (xy)^r = 1>. This is a very interesting family of groups which arises naturally in many contexts in number theory and topology.<br />
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6a. Prove that T(2,2,n) is isomorphic to the dihedral group D_n.<br />
6b. Prove that T(2,3,3) is finite and compute its order. Is it isomorphic to a finite group we've discussed in class?<br />
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(It is a beautiful fact, but outside the scope of this course, that T(p,q,r) is finite if and only if 1/p + 1/q + 1/r > 1.)<br />
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7. Let G be a subgroup of GL_n(R) which contains SL_n(R). Prove that G is a normal subgroup of GL_n(R).<br />
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8. Let G be a group and let H_1 and H_2 be subgroups of G of finite index (that is, the number of cosets in G/H_i is finite, i = 1,2.) Prove that H_1 intersect H_2 also has finite index in G.<br />
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9. Recall that the commutator [x,y] is x y x^{-1} y^{-1}. If G is a group, the commutator subgroup of G, denoted G' or [G,G], is the subgroup of G generated by all commutators [x,y] for x,y in G.<br />
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9a. Show that G' is a normal subgroup of G.<br />
9b. Show that G/G' is an abelian group.<br />
9c. Show that if f: G -> A is a homomorphism, with A abelian, then G' is contained in ker f. Conclude that f factors through a homomorphism G/G' -> A.<br />
9d. Show that G' = G if and only if G has no nontrivial abelian quotient. In this case, we say G is ''perfect''.<br />
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10. Under what conditions on (p,q,r) is the triangle group T(p,q,r) perfect?<br />
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==HOMEWORK 4 (due Oct 16)==<br />
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1. Let X be a set with a transitive action of a group G, which we think of as a homomorphism f: G -> Sym(X). Let H_x be the stabilizer of an element x of X.<br />
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1a. If x' is another element of X, show that H_{x"} and H_x are conjugate subgroups. (Hint: this is not true without the hypothesis of transitivity, so make sure to use it!)<br />
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1b. Show that the kernel of f is the intersection of all the conjugates of H_x in G. (This subgroup is called the ''normal core'' of H_x.)<br />
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2. This week, a certified mathematical grown-up asked on MathOverflow what the index-3 subgroups of SL_2(Z/pZ) were. Let's do it ourselves for homework. Let p be a prime, and suppose H is an index-3 subgroup of SL_2(Z/pZ). I do not claim that the proof sketched below is necessarily the easiest, but it allows us to cover some ground that we won't get to do in lecture.<br />
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2a. Using the coset action, show that the normal core of H is the kernel of a homomorphism f from SL_2(Z/pZ) to S_3.<br />
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2b. A ''unipotent matrix'' is a matrix A such that (A-I)^n = 0 for some positive integer n. Prove that a unipotent element of SL_2(Z/pZ) has order dividing p.<br />
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2c. You may use the fact that unipotent matrices generate SL_2(Z/pZ). Given this fact, show that f must be trivial when p > 3.<br />
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2d. Using the above, show that SL_2(Z/pZ) has no index-3 subgroups when p > 3.<br />
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2e. (extra credit) What can you say about the minimal index of a proper subgroup of SL_2(Z/pZ)?<br />
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3. Let H be a subgroup of G of index 2. Prove that H is normal.<br />
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4. We proved two theorems in class: Lagrange's theorem, which says that the order of a subgroup of G divides |G|, and Cauchy's theorem, which says that if p divides |G| then there exists a subgroup of G of order p. Cauchy's theorem is a kind of partial converse to Lagrange's theorem, but the full converse is false. Give an example of a finite group G and a divisor n of |G| such that you can prove there is no subgroup of G of order n.<br />
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5. Let X be the set of lines in F_3^2 (here a line means a 1-dimensional linear subspace) so that |X| = 4. Let G be the projective linear group PGL_2(F_3), so that G acts on X via its action on F_3^2. This action can be thought of as a homomorphism from G to S_4. Show that this homomorphism is an ISOMORPHISM from PGL_2(F_3) to S_4.<br />
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6. The group S_3 has only three conjugacy classes. Prove that a finite group with at most three conjugacy classes has order at most 6.<br />
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7. Let X be the set of ordered triples of elements of {1,..,n}, for some n >= 3. Then S_n acts on X. How many orbits are there? (Hint: the answer does not depend on n.)<br />
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8. Suppose that G and H are groups and f: G -> H is a homomorphism. Recall the definition of the abelianization G^ab from the previous problem set.<br />
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8a. Show that the composition G -> H -> H^ab factors through a unique homomorphism G^ab -> H^ab, which we denote f^ab.<br />
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8b. Show that if f: G -> H and g: H -> Q are homomorphisms, then f^ab o g^ab = (f o g)^ab.<br />
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(For those reading MacLane, this constitutes a proof that abelianization is a '''functor''' from the category of groups to the category of abelian groups.)<br />
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==HOMEWORK 5 (due Oct 23)==<br />
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1. Show that there are only two different isomorphism classes of groups of order 14. (Hint: prove that such a group must be a semidirect product of a group of order 7 by a group of order 2.)<br />
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2. (variant of Burnside's Lemma) Let G be a finite group and H a normal subgroup. Suppose G acts on a finite set X. <br />
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2a. Show that if x and y lie in the same H-orbit, and g is an element of G, then gx and gy also lie in the same H-orbit. Thus, G/H has a natural action on the set of H-orbits X/H.<br />
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2b. Show that the average number of fixed points in X of an element of the coset gH is the same as the number of fixed points of the element gH of G/H in its action on X/H.<br />
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3. A ''central extension'' of a group G by an abelian group A is a group E, together with a surjective homomorphism E -> G whose kernel is central in E and is isomorphic to A.<br />
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3a. Show that the quaternion group is a central extension of (Z/2Z)^2 by Z/2Z.<br />
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3b. Give an example of another nonabelian group which is a central extension of (Z/2Z)^2 by Z/2Z and which is not isomorphic to the quaternion group.<br />
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4. Prove that a central extension of an abelian group is nilpotent.<br />
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5. Give two different composition series for S_4 and show that they have the same composition factors.<br />
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6. Suppose that A is a finitely generated abelian group, and B is a subgroup of A.<br />
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6a. Prove that B is finitely generated and that rank(B) <= rank(A).<br />
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6b. Prove that rank(A/B) = rank(A) - rank(B).<br />
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7. For this problem, you may use the fact (which I should have asked on homework earlier) that if H_1 and H_2 are subgroups of G, then<br />
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[G: H_1 intersect H_2] <= [G:H_1][G:H_2].<br />
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Prove that if G is a simple group, d is an integer greater than 1, and S is a subgroup of index d in G, then |G| <= d^d.<br />
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Conclude that, in particular, A_5 has no subgroup of index 3. On the other hand, exhibit a subgroup of A_5 of index 5.<br />
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(Note: d^d is not a sharp bound; can you get a bound of d factorial instead?)<br />
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<!--(This argument shows that a solvable subgroup of A_n has size at most about (n-1)!, but this is far from sharp; e.g. I found a citation to a 1967 paper of Dixon which shows that a solvable subgroup of A_n has size at most k^n where k is around 2.88.)--><br />
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8. Give an example of an abelian group A with a subgroup B such that B is not a direct summand of A.<br />
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9. Let A be the subgroup of Z^2 generated by the vector (1,0). Let B be the subgroup of Z^2 generated by the vector (3,3). Show that A and B are isomorphic to each other, but Z^2 / A is not isomorphic to Z^2 / B.<br />
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==HOMEWORK 6 (due Oct 30)==<br />
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1. In this problem, you will construct a functor F from the category of finite sets to the category of complex vector spaces. I'll tell you what it does on objects: if X is a finite set, then F(X) is the vector space spanned by a set of basis elements {e_x}_{x in X}, so that dim F(X) = |X|. Your job: complete the definition of F by describing the map from Mor_{Finite Sets}(X,Y) to Mor_{Vector Spaces}(F(X),F(Y)), and showing that your map respects composition.<br />
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2. If (V, rho) is a representation of a group G, denote by V^G the space of vectors in V such that gv = v for all g in G; these are called _invariant_ vectors. We saw that the permutation representation of S_3 has a one-dimensional space of invariants. Prove that if X is a finite set with G-action, and V_X the corresponding permutation representation of G, then<br />
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dim V_X^G = number of orbits of X.<br />
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3. (Invariants are functorial) Let G be a group. Then there is a category called C[G]-Mod whose objects are complex representations of G (i.e. pairs (V,rho) where V is a complex vector space and rho is a homomorphism from G to GL(V)). The morphisms in this category are the ones described in class: Mor((V,rho),(W,psi)) is the set of linear maps f from V to W such that<br />
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f(rho(g)(v)) = psi(g)(f(v))<br />
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for all g in G and all v in V.<br />
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Show that there is a functor H_0 from the category of representations of G to the category of vector spaces whose action on objects is given by<br />
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H_0((V) = V^G.<br />
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(In other words: your job again is to explain how to associate to a morphism from (V,rho) to (W,phi) a morphism from V^G to W^G, in a way that's compatible with composition.)<br />
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<!--4. The group PSL_2(Z) has a presentation of the form [U,V|U^2 = V^3 = 1]. Thus, an n-dimensional representation of PSL_2(Z) is precisely a pair of n x n matrices (rho(U),rho(V)), such that rho(U)^2 = rho(V)^3 = 1. Prove that for every n > 1 there are uncountably many isomorphism classes of n-dimensional representations of PSL_2(Z). (Extra credit: show that there are uncountably many isomorphism classes of _irreducible_ representations.)<br />
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An amazing theorem of Margulis (superrigidity) shows that SL_2(Z) is very special among arithmetic groups in this way. For instance, there are only _finitely_ many isomorphism classes of n-dimensional representations of PSL_d(Z) for any d > 2; those groups are "rigid."--><br />
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5. Let G be a finite group, let X be a set with G-action, and let chi be the character of the corresponding permutation representation. Show that, for each g in G, chi(g) is the number of elements of X fixed by g. (I am actually going to prove this in class.)<br />
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6. Let V be the space of homogeneous degree-3 polynomials in variables x_1, x_2, x_3; then S_3 acts on V by permutation of the three variables. Describe the decomposition of V into irreducible representations of S_3.<br />
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7a. Show that if chi is the character of a representation of a finite group G, then chi(g^{-1}) is the complex conjugate of chi.<br />
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7b. As a consequence, show that if g is conjugate to g^{-1} for all g in G, then chi(g) is real.<br />
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7c. The groups D_p and S_n have the property that g is conjugate to g^{-1} for _every_ g in G, so that every character is a real-valued function on G. Give an example of a group G where this is not the case, and give an example of a representation of G whose character takes non-real values.<br />
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==HOMEWORK 7 (due Nov 6)==<br />
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1. Let G be a finite group, let X be a finite set with G-action, and let chi be the character of the corresponding permutation representation. Show that, for each g in G, chi(g) is the number of elements of X fixed by g. (I am actually going to prove this in class.)<br />
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2a. Show that if chi is the character of a finite-dimensional representation of a finite group G, then chi(g^{-1}) is the complex conjugate of chi(g) for all g.<br />
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2b. As a consequence, show that if g is conjugate to g^{-1} for all g in G, then chi(g) is real.<br />
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2c. The groups D_p and S_n have the property that g is conjugate to g^{-1} for _every_ g in G, so that every character is a real-valued function on G. Give an example of a group G where this is not the case, and give an example of a representation of G whose character takes non-real values.<br />
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3. Let V and W be complex vector spaces. Let f be a linear transformation of V, and g a linear transformation of W. <br />
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3a. Show that there is a unique linear transformation F satisfying<br />
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F(v tensor w) = f(v) tensor g(w)<br />
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for all v in V and all w in W. We denote this transformation by f tensor g.<br />
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3b. Suppose V and W are finite dimensional. Show that Trace(f tensor g) = Trace(f) Trace(g). What does this say when f and g are both the identity transformation?<br />
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4. Let V be a vector space. We define Sym^2 V to be the quotient of V tensor V by the subspace generated by all elements of the form<br />
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v tensor w - w tensor v<br />
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for v,w in V.<br />
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Suppose dim V = n. What is dim Sym^2 V?<br />
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5. Show that Z/pZ tensor Z/qZ is zero when p and q are distinct primes.<br />
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==HOMEWORK 8 (due Nov 13)==<br />
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1. (Schur's lemma can be false over non-algebraically closed fields) Let G be Z/3Z and let g be a generator of G. Then G acts by cyclic permutations on the set {1,2,3}; let P be the corresponding permutation representation on the REAL (not complex!) vector space generated by e_1,e_2,e_3.<br />
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1a. Show that P breaks up as the direct sum of the trivial representation and a 2-dimensional representation V which is irreducible. (Remember, we are not working over the complex numbers, so you cannot use the criterion for irreducibility in terms of the character that we worked out in class for the complex case; you need to prove directly that V has no nontrivial proper subrepresentation.)<br />
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1b. Show that the map from V to V induced by g is G-equivariant, but is neither zero nor a scalar multiplication.<br />
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2. If V_1 and V_2 are representations (not necessarily irreducible!) of G, show that <chi_V_1, chi V_2> is a non-negative integer.<br />
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3. When a formula always yields a non-negative integer, it is often because it is secretly a formula for either the cardinality of some finite set or the dimension of some finite-dimensional vector space. In this case, show that<br />
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<chi_{V_1}, chi_{V_2}> = dim_C Hom_G(V_1,V_2)<br />
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where we recall that Hom_G(V_1,G_2) refers to the space of G-equivariant homomorphisms from V_1 to V_2.<br />
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4. If A is an abelian group, show that every irreducible representation of A is 1-dimensional. (Hint: show that there is a vector which is an eigenvector for every element of A!)<br />
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5. Let G be the group of order 20 which is the semidirect product of A by H, where A is Z/5Z and H is (Z/5Z)^* with its natural action on A. We will work out all the irreducible representations of A. <br />
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5a. First of all, consider the 5-dimensional permutation representation of G acting by left multiplication on the cosets of H. Show that this representation decomposes as a direct sum of the trivial representation and an irreducible 4-dimensional representation. (Hint: it is probably easier to use the character of this representation than to prove irreducibility directly.) <br />
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5b. Now show that there are 4 1-dimensional representations of G which factor through the quotient G/A (i.e. such that rho(A) is the identity.)<br />
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5c. Using the decomposition of the regular representation, show that the irreducible representations of G you have constructed are the only ones.<br />
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==HOMEWORK 9 (due Nov 20)==<br />
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0. Find all ring homomorphisms from Z to Z/30Z. [Warning: there are 8 of them, not 30.]<br />
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1. If R is a ring (not necessarily with 1) such that a^2 = a for every a in R, R is called a Boolean ring.<br />
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1a. Show that any nonzero Boolean ring has characteristic 2. (In other words, show that a+a=0 for all a in R.)<br />
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1b. Show that every Boolean ring is commutative.<br />
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2. Let phi : R -> S be a homomorphism of commutative rings with 1.<br />
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2a. If I is an ideal of S, show that phi^(-1) (I) is an ideal of R.<br />
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2b. If P is a prime ideal of S, show that phi^(-1) (P) is either R or a prime ideal of R.<br />
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2c. If M is a maximal ideal of S and phi is surjective, show that phi^(-1) (M) is a maximal ideal of R.<br />
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2d. Find a homomorphism phi : R -> S and a maximal ideal M of S such that phi^(-1) (M) is not a maximal ideal of R.<br />
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3. Let Nil(A) denote the set of nilpotent elements of a ring A, and let R be a commutative ring. In class it was observed that Nil(R) is an ideal of R.<br />
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3a. Find (a generator of) Nil(Z/720Z).<br />
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3b. Show that Nil(R/Nil(R)) = 0.<br />
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3c. If x is in Nil(R) and u is a unit, show that u+x is a unit.<br />
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3d. Show that Nil(S), where S is the 2x2 matrices with real coefficients, is not an ideal of S.<br />
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4a. If M is an R-module and I is an ideal of R such that im = 0 for all i in I and m in M, show that M carries the structure of an R/I-module.<br />
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4b. If S is a commutative ring with 1 and J is a maximal ideal of S, show that J/J^2 is a vector space over S/J.<br />
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5. A "divisible element" x of a Z-module is one such that for each positive integer k, there exists a y such that ky=x. Show that a free Z-module cannot contain a nonzero divisible element.<br />
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6. Prove that the direct sum of any collection of free R-modules is also a free R-module.<br />
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7. In this problem you will prove that the module M given by the direct product of copies of Z, indexed by the positive integers, is not a free Z-module. Therefore, assume that M is a free Z-module with basis B, and let N be the direct sum of copies of Z, as a submodule of M.<br />
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7a. Show there is a countable subset B1 of B such that N is contained in the submodule N1 generated by B1. [Hint: N is countable.]<br />
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7b. Show that M/N1 is a free Z-module.<br />
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7c. Let S = {(b1, b2, b3, ...) : b_i = i! or -i!}. Show that there is some s in S which is not in N1. [Hint: S is uncountable.]<br />
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7d. For any s in S but not N1, show that for each positive integer k, there exists some m in M with s-km in N1. Conclude that s+N1 is a divisible element of M/N1.<br />
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7e. Use problem 5 to obtain a contradiction.<br />
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8. In class I said "rank" is not always well-defined for R-modules if R is not commutative. Here is an example demonstrating this assertion. Let M be the countably infinite direct product of copies of Z as in exercise 7, and let R be the endomorphism ring of M. Define phi_1(a1,a2,a3,a4,...) = (a1,a3,a5,...) and phi_2(a1,a2,a3,a4,...) = (a2,a4,a6,...).<br />
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8a. Show that {phi_1, phi_2} is a free basis for R, as a left R-module. [Hint: let psi_1(a1,a2,a3,...) = (a1,0,a2,0,...) and psi_2(a1,a2,a3,...) = (0,a1,0,a2,...). Compute the products of the phi_i and psi_j in either order, and use the result to show that the phi_i are a basis.]<br />
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8b. Show that R is isomorphic to R^n for every positive integer n.<br />
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==HOMEWORK 10 (due Dec 4)==<br />
<br />
1. '''How algebraists do calculus'''. The ''ring of dual numbers'' is the ring R = C[e]/(e^2). We think of e as an infinitesimal number; that is, something so small that its square is 0. This algebraic notion of infinitesimal is the basis for the algebraic notion of differentiation, which works as follows. Let f be a polynomial in R[x]. Show that there is a unique polynomial g in C[x] such that<br />
<br />
f(x+e) - f(x) = eg(x)<br />
<br />
and that in fact g is the derivative of f. In other words, the usual definition of "derivative" works just fine in this context, without any use of the notion of limit!<br />
<br />
2. Write down all the ideals of C[e]/e^2. (There are three.)<br />
<br />
3. Show that any right ideal of the ring M_2(Q) of 2x2 matrices is an ideal of the form I_V described in class for some subspace V of Q^2; that is, it consists precisely of those linear transformations whose image is contained in V.<br />
<br />
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4. An ''idempotent'' element of a ring is an element e satisfying e^2 = e.<br />
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4a. If e is an idempotent, show that 1-e is also an idempotent.<br />
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4b. Suppose R has a unit (we are actually always supposing this, but I just want to remind you of it for this problem) and suppose e is a ''central'' idempotent, i.e. an idempotent which commutes with all elements of R. Show that eR and (1-e)R are both two-sided ideals. Moreover, show that eR and (1-e)R are also subrings of R (with the warning that the identity in eR is not 1, which is not contained in eR, but rather e.)<br />
<br />
4c. Show that the map f_e: R -> eR defined by f_e(r) = er is a ring homomorphism. Finally show that the ring homomorphism<br />
<br />
f_{e} x f_{1-e}: R -> eR x (1-e)R <br />
<br />
is actually an isomorphism of rings. In other words, the presence of a central idempotent induces a decomposition a ring as a direct product.<br />
<br />
<br />
5. The ''center'' of a ring R is the subring of elements commuting with every element of R.<br />
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5a. Show that the center of M_n(Q) is the ring of scalar matrices.<br />
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5b. Show that the center of the group ring C[G] is the set of elements sum f(g) g such that f(g) is a class function.<br />
<br />
<br />
6. '''How algebraists do differential equations.''' We define the ''Weyl algebra'' W of differential operators in one variable as follows. Multiplication by x is a linear transformation on the complex vector space V of all smooth functions on C. Differentiation, which we denote d, is also a linear transformation on V. So we let W be the subring of End(V) generated by x and d. We note that x and d satisfy the relation xd = dx - 1; in particular, the Weyl algebra is not commutative.<br />
<br />
6a. Describe the submodule of V which is annihilated by the element d-x in W.<br />
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6b. Show that, for any polynomial f, fd -df = -f'. (This is another way algebraists do calculus!) Use this fact to show that the center of the Weyl algebra is just the field of constants. (Remark: you will use in the course of this proof the fact that a polynomial whose derivative is 0 is constant -- this would not be true if we were working in positive characteristic, as the example of x^p considered as a polynomial over F_p demonstrates.)<br />
<br />
<br />
7. If M is a left module for a ring R, we denote by Ann(M) the set of elements r in R such that rm = 0 for all m in M. Show that Ann(M) is a two-sided ideal. <br />
<br />
<br />
8. Let M be a left module for R, and let m be an element of M. We denote by Ann(m) the set of elements r in R such that rm = 0.<br />
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8a. Show that Ann(m) is a left ideal, and show that the submodule Rm of M generated by m is isomorphic to the left module R/Ann(m).<br />
<br />
8c. Let V,W be as in question 6. Show that Ann(e^x) is the left ideal W(d-1).<br />
<br />
==HOMEWORK 11 (due Dec 13)==<br />
<br />
1. Let f: R -> S be a homomorphism of rings. As discussed in class, if M is a left R-module, then S tensor_R M is a left S-module, and if N is an S-module, then N can also be considered as an R-module by pullback of the action.<br />
<br />
Prove that if M is a finitely generated left R-module, then S tensor_R M is a finitely generated left S-module. On the other hand, give an example to show that N can be finitely generated as a left S-module but not finitely generated when considered as a left R-module.<br />
<br />
2. Recall from class that when V and W are complex vector spaces, W^dual tensor V is naturally isomorphic to the space of linear homomorphisms Hom(W,V).<br />
<br />
Now suppose that V and W are left C[G]-modules, i.e. complex representations of a group G. Show that W^dual tensor_C[G} V is the maximal G-invariant quotient of Hom(W,V); in other words, the quotient of Hom(W,V) by the subspace generated by all functions F of the form F(w) = f(gw) - gf(w) for some function f in Hom(W,V) and some g in G.<br />
<br />
3. '''How algebraists do logic.''' Why are Boolean rings called Boolean rings? Because they model some features of the Boolean arithmetic that governs propositional logic. Let A be a collection of propositions. We define two operations on A as follows. By a + b we mean the negation of a xor b, while by ab we mean "a or b."<br />
<br />
3a. It is not a priori obvious that these operations satisfy the ring axioms, but assume this for the moment. Show that a^2 = a for all a (so A is a boolean ring) and 2a = 0 for all a (so A has characteristic 2.)<br />
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3b. Suppose we have a consistent assignment of truth-values to the propositions in A. Show that the set of true statements of A forms an ideal of A.<br />
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3c. (Optional for people who like propositional logic) Show that a(b+c) = ab + ac with the given operations.<br />
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4. Let K be Q[x]/(x^3-2).<br />
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4a. Show that K is a field.<br />
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4b. Show that K tensor_Q K decomposes as the direct sum of two fields. What are they?<br />
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5. There was some discussion in class about when we can "go back and forth" between right modules and left modules, as is the case with group rings.<br />
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5a. If R is a ring, we denote R^op ("the opposite ring") to be the ring whose elements are those of R, but whose multiplication law x_opp is defined by s x_opp r = rs. Show that if M is a left R-module, then the map M x R^op -> M sending (x,r) to rx makes M into a right R^op module. In particular, if there is a RING ISOMORPHISM from R to R^op, then every left R-module can be thought of as a right R-module.<br />
<br />
5b. Verify that the map sending g to g^{-1} extends by linearity to give a ring isomorphism between C[G] and C[G]^op.<br />
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5c. Construct a ring isomorphism between M_n(Q) and M_n(Q)^op.<br />
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5d. Optional: Give an example of a ring which is not isomorphic to its opposite ring. (This is not so easy, I think! I found a MathOverflow thread with several examples, none of which came with a short proof of the non-isomorphism -- try your hand!)<br />
<br />
6. . Let R be a commutative ring with 1. If R has a unique maximal ideal M, R is called a "local ring".<br />
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6a. If R is a local ring, show that every element of R that is not in M is a unit.<br />
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6b. Conversely, if R is such that the set of nonunits forms an ideal I, show that R is a local ring with maximal ideal I.<br />
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7. The inclusion of S_3 into S_4 associated to the inclusion of {1,2,3} into {1,2,3,4} induces a ring homomorphism from C[S_3} to C{S_4]. Let V be a representation of C{S_3]. Then, as discussed in class, V tensor_C[S_3] C[S_4] is a representation of S_4, called the '''induced representation''' of V.<br />
<br />
7a. Show that dim_C (V tensor_C[S_3] C[S_4]) = 4 dim_C V.<br />
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7b. When V is the trivial representation of S_3, decompose (V tensor_C[S_3] C[S_4]) as a sum of irreducible representations of S_4.<br />
<br />
<!--==HOMEWORK PROBLEMS TO GIVE IN FUTURE SEMESTERS==<br />
<br />
Let f: C[x] tensor_C C[y] -> C[z] be the map of rings with f(x) = f(y) = z. Show that f(m tensor n) is not 0 for any pure tensor m tensor n, but f is not injective. (This is a warning not to try to check injectivity by checking that no pure tensors are killed by the map.)<br />
<br />
--></div>Dummithttps://hilbert.math.wisc.edu/wiki/index.php?title=741&diff=48107412012-12-22T10:30:17Z<p>Dummit: /* HOMEWORK 5 (due Oct 23) */</p>
<hr />
<div>'''Math 741'''<br />
<br />
Algebra<br />
<br />
Prof: [http://www.math.wisc.edu/~ellenber Jordan Ellenberg]<br />
<br />
Grader: Evan Dummit<br />
<br />
Ellenberg's office hours: Friday 3pm.<br />
<br />
Grader's office hours: Monday 4pm. Late homework may be given directly to the grader, along with either (i) the instructor's permission, or (ii) a polite request for mercy.<br />
<br />
This course, the first semester of the introductory graduate sequence in algebra, will cover the basic theory of groups, group actions, representation, linear and multilinear algebra, and the beginnings of ring theory.<br />
<br />
==SYLLABUS==<br />
<br />
In this space we will record the theorems and definitions we covered each week, which we can use as a list of notions you should be prepared to answer questions about on the Algebra qualifying exam. The material covered on the homework is also an excellent guide to the scope of the course. <br />
<br />
'''WEEK 1''': <br />
<br />
Definition of group. Associativity. Inverse. <br />
<br />
Examples of group: GL_n(R). GL_n(Z). Z/nZ. R. Z. R^*. The free group F_k on k generators. <br />
<br />
Homomorphisms. The homomorphisms from F_k to G are in bijection with G^k. Isomorphisms.<br />
<br />
'''WEEK 2''':<br />
<br />
The symmetric group (or permutation group) S_n on n letters. Cycle decomposition of a permutation. Order of a permutation. Thm: every element of a finite group has finite order. <br />
<br />
Subgroups. Left and right cosets. Lagrange's Theorem. Cyclic groups. The order of an element of a finite group is a divisor of the order of the group.<br />
<br />
The sign homomorphism S_n -> +-1.<br />
<br />
<br />
'''WEEK 3'''<br />
<br />
Normal subgroups. The quotient of a group by a normal subgroup. The first isomorphism theorem. Examples of S_n -> +-1 and S_4 -> S_3 with kernel V_4, the Klein 4-group.<br />
<br />
Centralizers and centers. Abelian groups. The center of SL_n(R) is either 1 or +-1.<br />
<br />
Groups with presentations. The infinite dihedral group <x,y | x^2 = 1, y^2 = 1>.<br />
<br />
'''WEEK 4'''<br />
<br />
More on groups with presentations.<br />
<br />
Second and third isomorphism theorems.<br />
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Semidirect products.<br />
<br />
'''WEEK 5'''<br />
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Group actions, orbits, and stabilizers.<br />
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Orbit-stabilizer theorem.<br />
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Cayley's theorem.<br />
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Cauchy's theorem.<br />
<br />
'''WEEK 6'''<br />
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Applications of orbit-stabilizer theorem (p-groups have nontrivial center, first Sylow theorem.)<br />
<br />
Classification of finite abelian groups and finitely generated abelian groups.<br />
<br />
Composition series and the Jordan-Holder theorem (which we state but don't prove.) <br />
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The difference between knowing the composition factors and knowing the group (e.g. all p-groups of the same order have the same composition factors.)<br />
<br />
'''WEEK 7'''<br />
<br />
Simplicity of A_n.<br />
<br />
Nilpotent groups (main example: the Heisenberg group)<br />
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Derived series and lower central series.<br />
<br />
Category theory 101: Definition of category and functor. Some examples. A group is a groupoid with one object.<br />
<br />
'''WEEK 8'''<br />
<br />
Introduction to representation theory.<br />
<br />
'''WEEK 10'''<br />
<br />
Ring theory 101: Rings, ring homomorphisms, ideals, isomorphism theorems. Examples: fields, Z, the Hamilton quaternions, matrix rings, rings of polynomials and formal power series, quadratic integer rings, group rings. Integral domains. Maximal and prime ideals. The nilradical.<br />
<br />
Module theory 101: Modules, module homomorphisms, submodules, isomorphism theorems. Noetherian modules and Zorn's lemma. Direct sums and direct products of arbitrary collections of modules.<br />
<br />
<br />
Below you will find a repository of homework problems. Note that some of these problems are taken from Lang's ''Algebra''.<br />
<br />
==HOMEWORK 1 (due Sep 20)==<br />
<br />
1. Suppose that H_1 and H_2 are subgroups of a group G. Prove that the intersection of H_1 and H_2 is a subgroup of G.<br />
<br />
2. Recall that S_3 (the symmetric group) is the group of permutations of the set {1..3}. List all the subgroups of S_3.<br />
<br />
3. We can define an equivalence relation on rational numbers by declaring two rational numbers to be equal whenever they differ by an integer. We denote the set of equivalence classes by Q/Z. The operation of addition makes Q/Z into a group.<br />
<br />
a) For each n, prove that Q/Z has a subgroup of order n.<br />
<br />
b) Prove that Q/Z is a ''divisible'' group: that is, if x is an element of Q/Z and n is an integer, there exists an element y of Q/Z such that ny = x. (Note that we write the operation in this group as addition rather than multiplication, which is why we write ny for the n-fold product of y with itself rather than y^n)<br />
<br />
c) Prove that Q/Z is not finitely generated. (Hint: prove that if x_1, .. x_d is a finite subset of Q/Z, the subgroup of Q/Z generated by x_1, ... x_d is finite.)<br />
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4. Let F_2 be the free group on two generators x,y. Prove that there is a automorphism of F_2 which sends x to xyx and y to xy, and prove that this automorphism is unique.<br />
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5. Let H be a subgroup of G, and let N_G(H), the ''normalizer'' of H in G, be the set of elements g in G satsifying g H g^{-1} = H. Prove that N_G(H) is a subgroup of G.<br />
<br />
6. Let T be the subgroup of GL_2(R) consisting of diagonal matrices. This is an example of a "Cartan subgroup," or "torus" (whence the notation T.) Describe the normalizer N(T) of T in GL_2(R). Prove that there is a homomorphism from N(T) to Z/2Z whose kernel is precisely T. <br />
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7. Prove that the group of inner automorphisms of a group is normal in the full automorphism group Aut(G).<br />
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8. Let H and H' be subgroups of a group G, and let x be an element of G. We denote by HxH' the set of elements of the form hxh', with h in H and h' in H', and we call HxH a ''double coset'' of the pair (H,H').<br />
<br />
a) Show that G decomposes as a disjoint union of double cosets of (H,H').<br />
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b) Let G be GL_2(R) and let B be the subgroup of upper triangular matrices. Prove that G decomposes into exactly two double cosets of (B,B).<br />
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c) Let G be the symmetric group S_n, and let H be the subgroup of permutations which fix 1. Describe the double coset decomposition of S_n into double cosets of (H,H).<br />
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==HOMEWORK 2 (due Sep 27)==<br />
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1. Prove that the symmetric group S_n is generated by transpositions. Hint: proceed by induction on n. Let H be the subgroup of S_n consisting of permutations fixing 1. Observe that H is isomorphic to S_{n-1}, so that H is generated by transpositions. Explain why it suffices to show that, for any g in S_n, the coset gH contains a transposition. Then explain why this is the case.<br />
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2. Given that the symmetric group S_n is generated by transpositions, show that there are no nontrivial homomorphisms from S_n to Z/pZ for any odd prime p.<br />
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3. Compute the centralizer of the triple flip (12)(34)(56) in S_6. (Note: this is the kind of problem that a computer algebra package like sage is very good at, and I encourage you to learn to use sage well enough to check your answer.)<br />
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4. Warning: a normal subgroup of a normal subgroup need not be normal! Let H be the subgroup of S_4 generated by (12)(34). Then H is a subgroup of the Klein 4-group V_4, which we have already shown is normal in S_4. Show that H is normal in V_4, but H is not normal in S_4.<br />
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5. The '''quaternion group''' Q is a non-abelian finite group of order 8. Its elements are 1, -1, i, -i, j, -j, k, -k; it satisfies g^2 = -1 for all g except +-1, and ijk = 1. (Note that the notation "-i" means "the product of i with -1" and -1 is understood to be central in Q.) Prove that every subgroup of Q is normal.<br />
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6. (finishing example done in class) Let Gamma be the group F<x,y> / (x^2 = 1, y^2 = 1). (Remember, this means the quotient of the free group by the group generated by all conjugates of x^2 and y^2.) Prove that Gamma is infinite. Prove that Gamma has a subgroup R generated by xy isomorphic to Z, that R is normal, and that the quotient Gamma/R is isomorphic to Z/2Z. <br />
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7. Suppose a group G has the property that every element in g satisfies g^2 = 1. Show that G must be abelian.<br />
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==HOMEWORK 3 (due Oct 4)==<br />
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1. Multiplication by -1 is an automorphism of Z/nZ. Let a be the homomorphism Z/2Z -> Aut(Z/nZ) which sends the nontrivial element of Z/2Z to multiplication by -1. Then we have a semidirect product of Z/pZ by Z/2Z as defined in class. This group of order 2n is called a ''dihedral group.'' and is denoted D_n (or sometimes D_{2n}).<br />
<br />
1a. Compute the center of D_n. (Note that the answer depends on n!)<br />
1b. Show that all involutions of D_n are conjugate to each other if and only if n is odd.<br />
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2. (More on direct products that we didn't do in class.) Suppose that G is a semidirect product of N with H. Suppose furthermore that H is also normal in G. Prove that G is a direct product of N with H.<br />
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3. Let Q be the quaternion group from last week's homework. Show that Q does not decompose as a semidirect product N \rtimes H, apart form the trivial cases N = {e} and N = Q.<br />
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4. The ''affine linear group'' of degree n is the group of transformations from R^n to R^n of the form x -> Ax + b, where A is a matrix in GL_n(R) and b is a vector in R^n. Prove that the affine linear group is a semidirect product R^n \rtimes GL_n.<br />
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5. The ''ordinary triangle group'' T(p,q,r) is the group with presentation <x,y | x^p = y^q = (xy)^r = 1>. This is a very interesting family of groups which arises naturally in many contexts in number theory and topology.<br />
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6a. Prove that T(2,2,n) is isomorphic to the dihedral group D_n.<br />
6b. Prove that T(2,3,3) is finite and compute its order. Is it isomorphic to a finite group we've discussed in class?<br />
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(It is a beautiful fact, but outside the scope of this course, that T(p,q,r) is finite if and only if 1/p + 1/q + 1/r > 1.)<br />
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7. Let G be a subgroup of GL_n(R) which contains SL_n(R). Prove that G is a normal subgroup of GL_n(R).<br />
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8. Let G be a group and let H_1 and H_2 be subgroups of G of finite index (that is, the number of cosets in G/H_i is finite, i = 1,2.) Prove that H_1 intersect H_2 also has finite index in G.<br />
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9. Recall that the commutator [x,y] is x y x^{-1} y^{-1}. If G is a group, the commutator subgroup of G, denoted G' or [G,G], is the subgroup of G generated by all commutators [x,y] for x,y in G.<br />
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9a. Show that G' is a normal subgroup of G.<br />
9b. Show that G/G' is an abelian group.<br />
9c. Show that if f: G -> A is a homomorphism, with A abelian, then G' is contained in ker f. Conclude that f factors through a homomorphism G/G' -> A.<br />
9d. Show that G' = G if and only if G has no nontrivial abelian quotient. In this case, we say G is ''perfect''.<br />
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10. Under what conditions on (p,q,r) is the triangle group T(p,q,r) perfect?<br />
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==HOMEWORK 4 (due Oct 16)==<br />
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1. Let X be a set with a transitive action of a group G, which we think of as a homomorphism f: G -> Sym(X). Let H_x be the stabilizer of an element x of X.<br />
<br />
1a. If x' is another element of X, show that H_{x"} and H_x are conjugate subgroups. (Hint: this is not true without the hypothesis of transitivity, so make sure to use it!)<br />
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1b. Show that the kernel of f is the intersection of all the conjugates of H_x in G. (This subgroup is called the ''normal core'' of H_x.)<br />
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2. This week, a certified mathematical grown-up asked on MathOverflow what the index-3 subgroups of SL_2(Z/pZ) were. Let's do it ourselves for homework. Let p be a prime, and suppose H is an index-3 subgroup of SL_2(Z/pZ). I do not claim that the proof sketched below is necessarily the easiest, but it allows us to cover some ground that we won't get to do in lecture.<br />
<br />
2a. Using the coset action, show that the normal core of H is the kernel of a homomorphism f from SL_2(Z/pZ) to S_3.<br />
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2b. A ''unipotent matrix'' is a matrix A such that (A-I)^n = 0 for some positive integer n. Prove that a unipotent element of SL_2(Z/pZ) has order dividing p.<br />
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2c. You may use the fact that unipotent matrices generate SL_2(Z/pZ). Given this fact, show that f must be trivial when p > 3.<br />
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2d. Using the above, show that SL_2(Z/pZ) has no index-3 subgroups when p > 3.<br />
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2e. (extra credit) What can you say about the minimal index of a proper subgroup of SL_2(Z/pZ)?<br />
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3. Let H be a subgroup of G of index 2. Prove that H is normal.<br />
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4. We proved two theorems in class: Lagrange's theorem, which says that the order of a subgroup of G divides |G|, and Cauchy's theorem, which says that if p divides |G| then there exists a subgroup of G of order p. Cauchy's theorem is a kind of partial converse to Lagrange's theorem, but the full converse is false. Give an example of a finite group G and a divisor n of |G| such that you can prove there is no subgroup of G of order n.<br />
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5. Let X be the set of lines in F_3^2 (here a line means a 1-dimensional linear subspace) so that |X| = 4. Let G be the projective linear group PGL_2(F_3), so that G acts on X via its action on F_3^2. This action can be thought of as a homomorphism from G to S_4. Show that this homomorphism is an ISOMORPHISM from PGL_2(F_3) to S_4.<br />
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6. The group S_3 has only three conjugacy classes. Prove that a finite group with at most three conjugacy classes has order at most 6.<br />
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7. Let X be the set of ordered triples of elements of {1,..,n}, for some n >= 3. Then S_n acts on X. How many orbits are there? (Hint: the answer does not depend on n.)<br />
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8. Suppose that G and H are groups and f: G -> H is a homomorphism. Recall the definition of the abelianization G^ab from the previous problem set.<br />
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8a. Show that the composition G -> H -> H^ab factors through a unique homomorphism G^ab -> H^ab, which we denote f^ab.<br />
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8b. Show that if f: G -> H and g: H -> Q are homomorphisms, then f^ab o g^ab = (f o g)^ab.<br />
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(For those reading MacLane, this constitutes a proof that abelianization is a '''functor''' from the category of groups to the category of abelian groups.)<br />
<br />
==HOMEWORK 5 (due Oct 23)==<br />
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1. Show that there are only two different isomorphism classes of groups of order 14. (Hint: prove that such a group must be a semidirect product of a group of order 7 by a group of order 2.)<br />
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2. (variant of Burnside's Lemma) Let G be a finite group and H a normal subgroup. Suppose G acts on a finite set X. <br />
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2a. Show that if x and y lie in the same H-orbit, and g is an element of G, then gx and gy also lie in the same H-orbit. Thus, G/H has a natural action on the set of H-orbits X/H.<br />
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2b. Show that the average number of fixed points in X of an element of the coset gH is the same as the number of fixed points of the element gH of G/H in its action on X/H.<br />
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3. A ''central extension'' of a group G by an abelian group A is a group E, together with a surjective homomorphism E -> G whose kernel is central in E and is isomorphic to A.<br />
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3a. Show that the quaternion group is a central extension of (Z/2Z)^2 by Z/2Z.<br />
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3b. Give an example of another nonabelian group which is a central extension of (Z/2Z)^2 by Z/2Z and which is not isomorphic to the quaternion group.<br />
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4. Prove that a central extension of an abelian group is nilpotent.<br />
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5. Give two different composition series for S_4 and show that they have the same composition factors.<br />
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6. Suppose that A is a finitely generated abelian group, and B is a subgroup of A.<br />
6a. Prove that B is finitely generated and that rank(B) <= rank(A).<br />
6b. Prove that rank(A/B) = rank(A) - rank(B).<br />
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7. For this problem, you may use the fact (which I should have asked on homework earlier) that if H_1 and H_2 are subgroups of G, then<br />
<br />
[G: H_1 intersect H_2] <= [G:H_1][G:H_2].<br />
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Prove that if G is a simple group, d is an integer greater than 1, and S is a subgroup of index d in G, then |G| <= d^d.<br />
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Conclude that, in particular, A_5 has no subgroup of index 3. On the other hand, exhibit a subgroup of A_5 of index 5.<br />
<br />
(Note: d^d is not a sharp bound; can you get a bound of d factorial instead?)<br />
<br />
<!--(This argument shows that a solvable subgroup of A_n has size at most about (n-1)!, but this is far from sharp; e.g. I found a citation to a 1967 paper of Dixon which shows that a solvable subgroup of A_n has size at most k^n where k is around 2.88.)--><br />
<br />
8. Give an example of an abelian group A with a subgroup B such that B is not a direct summand of A.<br />
<br />
9. Let A be the subgroup of Z^2 generated by the vector (1,0). Let B be the subgroup of Z^2 generated by the vector (3,3). Show that A and B are isomorphic to each other, but Z^2 / A is not isomorphic to Z^2 / B.<br />
<br />
==HOMEWORK 6 (due Oct 30)==<br />
<br />
1. In this problem, you will construct a functor F from the category of finite sets to the category of complex vector spaces. I'll tell you what it does on objects: if X is a finite set, then F(X) is the vector space spanned by a set of basis elements {e_x}_{x in X}, so that dim F(X) = |X|. Your job: complete the definition of F by describing the map from Mor_{Finite Sets}(X,Y) to Mor_{Vector Spaces}(F(X),F(Y)), and showing that your map respects composition.<br />
<br />
2. If (V, rho) is a representation of a group G, denote by V^G the space of vectors in V such that gv = v for all g in G; these are called _invariant_ vectors. We saw that the permutation representation of S_3 has a one-dimensional space of invariants. Prove that if X is a finite set with G-action, and V_X the corresponding permutation representation of G, then<br />
<br />
dim V_X^G = number of orbits of X.<br />
<br />
3. (Invariants are functorial) Let G be a group. Then there is a category called C[G]-Mod whose objects are complex representations of G (i.e. pairs (V,rho) where V is a complex vector space and rho is a homomorphism from G to GL(V)). The morphisms in this category are the ones described in class: Mor((V,rho),(W,psi)) is the set of linear maps f from V to W such that<br />
<br />
f(rho(g)(v)) = psi(g)(f(v))<br />
<br />
for all g in G and all v in V.<br />
<br />
Show that there is a functor H_0 from the category of representations of G to the category of vector spaces whose action on objects is given by<br />
<br />
H_0((V) = V^G.<br />
<br />
(In other words: your job again is to explain how to associate to a morphism from (V,rho) to (W,phi) a morphism from V^G to W^G, in a way that's compatible with composition.)<br />
<br />
<!--4. The group PSL_2(Z) has a presentation of the form [U,V|U^2 = V^3 = 1]. Thus, an n-dimensional representation of PSL_2(Z) is precisely a pair of n x n matrices (rho(U),rho(V)), such that rho(U)^2 = rho(V)^3 = 1. Prove that for every n > 1 there are uncountably many isomorphism classes of n-dimensional representations of PSL_2(Z). (Extra credit: show that there are uncountably many isomorphism classes of _irreducible_ representations.)<br />
<br />
An amazing theorem of Margulis (superrigidity) shows that SL_2(Z) is very special among arithmetic groups in this way. For instance, there are only _finitely_ many isomorphism classes of n-dimensional representations of PSL_d(Z) for any d > 2; those groups are "rigid."--><br />
<br />
5. Let G be a finite group, let X be a set with G-action, and let chi be the character of the corresponding permutation representation. Show that, for each g in G, chi(g) is the number of elements of X fixed by g. (I am actually going to prove this in class.)<br />
<br />
6. Let V be the space of homogeneous degree-3 polynomials in variables x_1, x_2, x_3; then S_3 acts on V by permutation of the three variables. Describe the decomposition of V into irreducible representations of S_3.<br />
<br />
7a. Show that if chi is the character of a representation of a finite group G, then chi(g^{-1}) is the complex conjugate of chi.<br />
<br />
7b. As a consequence, show that if g is conjugate to g^{-1} for all g in G, then chi(g) is real.<br />
<br />
7c. The groups D_p and S_n have the property that g is conjugate to g^{-1} for _every_ g in G, so that every character is a real-valued function on G. Give an example of a group G where this is not the case, and give an example of a representation of G whose character takes non-real values.<br />
<br />
==HOMEWORK 7 (due Nov 6)==<br />
<br />
1. Let G be a finite group, let X be a finite set with G-action, and let chi be the character of the corresponding permutation representation. Show that, for each g in G, chi(g) is the number of elements of X fixed by g. (I am actually going to prove this in class.)<br />
<br />
2a. Show that if chi is the character of a finite-dimensional representation of a finite group G, then chi(g^{-1}) is the complex conjugate of chi(g) for all g.<br />
<br />
2b. As a consequence, show that if g is conjugate to g^{-1} for all g in G, then chi(g) is real.<br />
<br />
2c. The groups D_p and S_n have the property that g is conjugate to g^{-1} for _every_ g in G, so that every character is a real-valued function on G. Give an example of a group G where this is not the case, and give an example of a representation of G whose character takes non-real values.<br />
<br />
3. Let V and W be complex vector spaces. Let f be a linear transformation of V, and g a linear transformation of W. <br />
<br />
3a. Show that there is a unique linear transformation F satisfying<br />
<br />
F(v tensor w) = f(v) tensor g(w)<br />
<br />
for all v in V and all w in W. We denote this transformation by f tensor g.<br />
<br />
3b. Suppose V and W are finite dimensional. Show that Trace(f tensor g) = Trace(f) Trace(g). What does this say when f and g are both the identity transformation?<br />
<br />
4. Let V be a vector space. We define Sym^2 V to be the quotient of V tensor V by the subspace generated by all elements of the form<br />
<br />
v tensor w - w tensor v<br />
<br />
for v,w in V.<br />
<br />
Suppose dim V = n. What is dim Sym^2 V?<br />
<br />
5. Show that Z/pZ tensor Z/qZ is zero when p and q are distinct primes.<br />
<br />
==HOMEWORK 8 (due Nov 13)==<br />
<br />
1. (Schur's lemma can be false over non-algebraically closed fields) Let G be Z/3Z and let g be a generator of G. Then G acts by cyclic permutations on the set {1,2,3}; let P be the corresponding permutation representation on the REAL (not complex!) vector space generated by e_1,e_2,e_3.<br />
<br />
1a. Show that P breaks up as the direct sum of the trivial representation and a 2-dimensional representation V which is irreducible. (Remember, we are not working over the complex numbers, so you cannot use the criterion for irreducibility in terms of the character that we worked out in class for the complex case; you need to prove directly that V has no nontrivial proper subrepresentation.)<br />
<br />
1b. Show that the map from V to V induced by g is G-equivariant, but is neither zero nor a scalar multiplication.<br />
<br />
2. If V_1 and V_2 are representations (not necessarily irreducible!) of G, show that <chi_V_1, chi V_2> is a non-negative integer.<br />
<br />
3. When a formula always yields a non-negative integer, it is often because it is secretly a formula for either the cardinality of some finite set or the dimension of some finite-dimensional vector space. In this case, show that<br />
<br />
<chi_{V_1}, chi_{V_2}> = dim_C Hom_G(V_1,V_2)<br />
<br />
where we recall that Hom_G(V_1,G_2) refers to the space of G-equivariant homomorphisms from V_1 to V_2.<br />
<br />
4. If A is an abelian group, show that every irreducible representation of A is 1-dimensional. (Hint: show that there is a vector which is an eigenvector for every element of A!)<br />
<br />
5. Let G be the group of order 20 which is the semidirect product of A by H, where A is Z/5Z and H is (Z/5Z)^* with its natural action on A. We will work out all the irreducible representations of A. <br />
<br />
5a. First of all, consider the 5-dimensional permutation representation of G acting by left multiplication on the cosets of H. Show that this representation decomposes as a direct sum of the trivial representation and an irreducible 4-dimensional representation. (Hint: it is probably easier to use the character of this representation than to prove irreducibility directly.) <br />
<br />
5b. Now show that there are 4 1-dimensional representations of G which factor through the quotient G/A (i.e. such that rho(A) is the identity.)<br />
<br />
5c. Using the decomposition of the regular representation, show that the irreducible representations of G you have constructed are the only ones.<br />
<br />
==HOMEWORK 9 (due Nov 20)==<br />
<br />
0. Find all ring homomorphisms from Z to Z/30Z. [Warning: there are 8 of them, not 30.]<br />
<br />
<br />
1. If R is a ring (not necessarily with 1) such that a^2 = a for every a in R, R is called a Boolean ring.<br />
<br />
1a. Show that any nonzero Boolean ring has characteristic 2. (In other words, show that a+a=0 for all a in R.)<br />
<br />
1b. Show that every Boolean ring is commutative.<br />
<br />
<br />
2. Let phi : R -> S be a homomorphism of commutative rings with 1.<br />
<br />
2a. If I is an ideal of S, show that phi^(-1) (I) is an ideal of R.<br />
<br />
2b. If P is a prime ideal of S, show that phi^(-1) (P) is either R or a prime ideal of R.<br />
<br />
2c. If M is a maximal ideal of S and phi is surjective, show that phi^(-1) (M) is a maximal ideal of R.<br />
<br />
2d. Find a homomorphism phi : R -> S and a maximal ideal M of S such that phi^(-1) (M) is not a maximal ideal of R.<br />
<br />
<br />
3. Let Nil(A) denote the set of nilpotent elements of a ring A, and let R be a commutative ring. In class it was observed that Nil(R) is an ideal of R.<br />
<br />
3a. Find (a generator of) Nil(Z/720Z).<br />
<br />
3b. Show that Nil(R/Nil(R)) = 0.<br />
<br />
3c. If x is in Nil(R) and u is a unit, show that u+x is a unit.<br />
<br />
3d. Show that Nil(S), where S is the 2x2 matrices with real coefficients, is not an ideal of S.<br />
<br />
<br />
4a. If M is an R-module and I is an ideal of R such that im = 0 for all i in I and m in M, show that M carries the structure of an R/I-module.<br />
<br />
4b. If S is a commutative ring with 1 and J is a maximal ideal of S, show that J/J^2 is a vector space over S/J.<br />
<br />
<br />
5. A "divisible element" x of a Z-module is one such that for each positive integer k, there exists a y such that ky=x. Show that a free Z-module cannot contain a nonzero divisible element.<br />
<br />
<br />
6. Prove that the direct sum of any collection of free R-modules is also a free R-module.<br />
<br />
<br />
7. In this problem you will prove that the module M given by the direct product of copies of Z, indexed by the positive integers, is not a free Z-module. Therefore, assume that M is a free Z-module with basis B, and let N be the direct sum of copies of Z, as a submodule of M.<br />
<br />
7a. Show there is a countable subset B1 of B such that N is contained in the submodule N1 generated by B1. [Hint: N is countable.]<br />
<br />
7b. Show that M/N1 is a free Z-module.<br />
<br />
7c. Let S = {(b1, b2, b3, ...) : b_i = i! or -i!}. Show that there is some s in S which is not in N1. [Hint: S is uncountable.]<br />
<br />
7d. For any s in S but not N1, show that for each positive integer k, there exists some m in M with s-km in N1. Conclude that s+N1 is a divisible element of M/N1.<br />
<br />
7e. Use problem 5 to obtain a contradiction.<br />
<br />
<br />
8. In class I said "rank" is not always well-defined for R-modules if R is not commutative. Here is an example demonstrating this assertion. Let M be the countably infinite direct product of copies of Z as in exercise 7, and let R be the endomorphism ring of M. Define phi_1(a1,a2,a3,a4,...) = (a1,a3,a5,...) and phi_2(a1,a2,a3,a4,...) = (a2,a4,a6,...).<br />
<br />
8a. Show that {phi_1, phi_2} is a free basis for R, as a left R-module. [Hint: let psi_1(a1,a2,a3,...) = (a1,0,a2,0,...) and psi_2(a1,a2,a3,...) = (0,a1,0,a2,...). Compute the products of the phi_i and psi_j in either order, and use the result to show that the phi_i are a basis.]<br />
<br />
8b. Show that R is isomorphic to R^n for every positive integer n.<br />
<br />
<br />
==HOMEWORK 10 (due Dec 4)==<br />
<br />
1. '''How algebraists do calculus'''. The ''ring of dual numbers'' is the ring R = C[e]/(e^2). We think of e as an infinitesimal number; that is, something so small that its square is 0. This algebraic notion of infinitesimal is the basis for the algebraic notion of differentiation, which works as follows. Let f be a polynomial in R[x]. Show that there is a unique polynomial g in C[x] such that<br />
<br />
f(x+e) - f(x) = eg(x)<br />
<br />
and that in fact g is the derivative of f. In other words, the usual definition of "derivative" works just fine in this context, without any use of the notion of limit!<br />
<br />
2. Write down all the ideals of C[e]/e^2. (There are three.)<br />
<br />
3. Show that any right ideal of the ring M_2(Q) of 2x2 matrices is an ideal of the form I_V described in class for some subspace V of Q^2; that is, it consists precisely of those linear transformations whose image is contained in V.<br />
<br />
<br />
4. An ''idempotent'' element of a ring is an element e satisfying e^2 = e.<br />
<br />
4a. If e is an idempotent, show that 1-e is also an idempotent.<br />
<br />
4b. Suppose R has a unit (we are actually always supposing this, but I just want to remind you of it for this problem) and suppose e is a ''central'' idempotent, i.e. an idempotent which commutes with all elements of R. Show that eR and (1-e)R are both two-sided ideals. Moreover, show that eR and (1-e)R are also subrings of R (with the warning that the identity in eR is not 1, which is not contained in eR, but rather e.)<br />
<br />
4c. Show that the map f_e: R -> eR defined by f_e(r) = er is a ring homomorphism. Finally show that the ring homomorphism<br />
<br />
f_{e} x f_{1-e}: R -> eR x (1-e)R <br />
<br />
is actually an isomorphism of rings. In other words, the presence of a central idempotent induces a decomposition a ring as a direct product.<br />
<br />
<br />
5. The ''center'' of a ring R is the subring of elements commuting with every element of R.<br />
<br />
5a. Show that the center of M_n(Q) is the ring of scalar matrices.<br />
<br />
5b. Show that the center of the group ring C[G] is the set of elements sum f(g) g such that f(g) is a class function.<br />
<br />
<br />
6. '''How algebraists do differential equations.''' We define the ''Weyl algebra'' W of differential operators in one variable as follows. Multiplication by x is a linear transformation on the complex vector space V of all smooth functions on C. Differentiation, which we denote d, is also a linear transformation on V. So we let W be the subring of End(V) generated by x and d. We note that x and d satisfy the relation xd = dx - 1; in particular, the Weyl algebra is not commutative.<br />
<br />
6a. Describe the submodule of V which is annihilated by the element d-x in W.<br />
<br />
6b. Show that, for any polynomial f, fd -df = -f'. (This is another way algebraists do calculus!) Use this fact to show that the center of the Weyl algebra is just the field of constants. (Remark: you will use in the course of this proof the fact that a polynomial whose derivative is 0 is constant -- this would not be true if we were working in positive characteristic, as the example of x^p considered as a polynomial over F_p demonstrates.)<br />
<br />
<br />
7. If M is a left module for a ring R, we denote by Ann(M) the set of elements r in R such that rm = 0 for all m in M. Show that Ann(M) is a two-sided ideal. <br />
<br />
<br />
8. Let M be a left module for R, and let m be an element of M. We denote by Ann(m) the set of elements r in R such that rm = 0.<br />
<br />
8a. Show that Ann(m) is a left ideal, and show that the submodule Rm of M generated by m is isomorphic to the left module R/Ann(m).<br />
<br />
8c. Let V,W be as in question 6. Show that Ann(e^x) is the left ideal W(d-1).<br />
<br />
==HOMEWORK 11 (due Dec 13)==<br />
<br />
1. Let f: R -> S be a homomorphism of rings. As discussed in class, if M is a left R-module, then S tensor_R M is a left S-module, and if N is an S-module, then N can also be considered as an R-module by pullback of the action.<br />
<br />
Prove that if M is a finitely generated left R-module, then S tensor_R M is a finitely generated left S-module. On the other hand, give an example to show that N can be finitely generated as a left S-module but not finitely generated when considered as a left R-module.<br />
<br />
2. Recall from class that when V and W are complex vector spaces, W^dual tensor V is naturally isomorphic to the space of linear homomorphisms Hom(W,V).<br />
<br />
Now suppose that V and W are left C[G]-modules, i.e. complex representations of a group G. Show that W^dual tensor_C[G} V is the maximal G-invariant quotient of Hom(W,V); in other words, the quotient of Hom(W,V) by the subspace generated by all functions F of the form F(w) = f(gw) - gf(w) for some function f in Hom(W,V) and some g in G.<br />
<br />
3. '''How algebraists do logic.''' Why are Boolean rings called Boolean rings? Because they model some features of the Boolean arithmetic that governs propositional logic. Let A be a collection of propositions. We define two operations on A as follows. By a + b we mean the negation of a xor b, while by ab we mean "a or b."<br />
<br />
3a. It is not a priori obvious that these operations satisfy the ring axioms, but assume this for the moment. Show that a^2 = a for all a (so A is a boolean ring) and 2a = 0 for all a (so A has characteristic 2.)<br />
<br />
3b. Suppose we have a consistent assignment of truth-values to the propositions in A. Show that the set of true statements of A forms an ideal of A.<br />
<br />
3c. (Optional for people who like propositional logic) Show that a(b+c) = ab + ac with the given operations.<br />
<br />
<br />
4. Let K be Q[x]/(x^3-2).<br />
<br />
4a. Show that K is a field.<br />
<br />
4b. Show that K tensor_Q K decomposes as the direct sum of two fields. What are they?<br />
<br />
5. There was some discussion in class about when we can "go back and forth" between right modules and left modules, as is the case with group rings.<br />
<br />
5a. If R is a ring, we denote R^op ("the opposite ring") to be the ring whose elements are those of R, but whose multiplication law x_opp is defined by s x_opp r = rs. Show that if M is a left R-module, then the map M x R^op -> M sending (x,r) to rx makes M into a right R^op module. In particular, if there is a RING ISOMORPHISM from R to R^op, then every left R-module can be thought of as a right R-module.<br />
<br />
5b. Verify that the map sending g to g^{-1} extends by linearity to give a ring isomorphism between C[G] and C[G]^op.<br />
<br />
5c. Construct a ring isomorphism between M_n(Q) and M_n(Q)^op.<br />
<br />
5d. Optional: Give an example of a ring which is not isomorphic to its opposite ring. (This is not so easy, I think! I found a MathOverflow thread with several examples, none of which came with a short proof of the non-isomorphism -- try your hand!)<br />
<br />
6. . Let R be a commutative ring with 1. If R has a unique maximal ideal M, R is called a "local ring".<br />
<br />
6a. If R is a local ring, show that every element of R that is not in M is a unit.<br />
<br />
6b. Conversely, if R is such that the set of nonunits forms an ideal I, show that R is a local ring with maximal ideal I.<br />
<br />
7. The inclusion of S_3 into S_4 associated to the inclusion of {1,2,3} into {1,2,3,4} induces a ring homomorphism from C[S_3} to C{S_4]. Let V be a representation of C{S_3]. Then, as discussed in class, V tensor_C[S_3] C[S_4] is a representation of S_4, called the '''induced representation''' of V.<br />
<br />
7a. Show that dim_C (V tensor_C[S_3] C[S_4]) = 4 dim_C V.<br />
<br />
7b. When V is the trivial representation of S_3, decompose (V tensor_C[S_3] C[S_4]) as a sum of irreducible representations of S_4.<br />
<br />
<!--==HOMEWORK PROBLEMS TO GIVE IN FUTURE SEMESTERS==<br />
<br />
Let f: C[x] tensor_C C[y] -> C[z] be the map of rings with f(x) = f(y) = z. Show that f(m tensor n) is not 0 for any pure tensor m tensor n, but f is not injective. (This is a warning not to try to check injectivity by checking that no pure tensors are killed by the map.)<br />
<br />
--></div>Dummithttps://hilbert.math.wisc.edu/wiki/index.php?title=NTSGrad_Fall_2012&diff=4772NTSGrad Fall 20122012-12-04T04:37:47Z<p>Dummit: /* Fall 2012 Semester */</p>
<hr />
<div>= Number Theory – Representation Theory Graduate Student Seminar, University of Wisconsin–Madison =<br />
<br />
*'''When:''' Tuesdays, 2:30pm–3:30pm,<br />
*'''Where:''' Van Vleck B113<br />
<br />
The purpose of this seminar is to have a talk on each Tuesday by a graduate student to<br />
help orient ourselves for the [[NTS|Number Theory Seminar]] talk on the following Thursday.<br />
These talks should be aimed at beginning graduate students, and should try to <br />
explain some of the background, terminology, and ideas for the Thursday talk. <br />
<br />
== Fall 2012 Semester ==<br />
<br />
<center><br />
<br />
{| style="color:black; font-size:120%" border="0" cellpadding="14" cellspacing="5"<br />
|-<br />
| bgcolor="#D0D0D0" width="300" align="center"|'''Date'''<br />
| bgcolor="#F0A0A0" width="300" align="center"|'''Speaker'''<br />
| bgcolor="#BCD2EE" width="300" align="center"|'''Title (click to see abstract)'''<br />
|- <br />
| bgcolor="#E0E0E0"| Sep 11 (Tues.)<br />
| bgcolor="#F0B0B0"| Silas Johnson <br />
| bgcolor="#BCE2FE"|[[NTS Fall 2012/Abstracts#September 11 | <font color="black"><em>tba</em></font>]]<br />
|- <br />
| bgcolor="#E0E0E0"| Sep 18 (Tues.)<br />
| bgcolor="#F0B0B0"| Marton Hablicsek<br />
| bgcolor="#BCE2FE"|[[NTS Fall 2012/Abstracts#September 18 | <font color="black"><em>tba</em></font>]]<br />
|- <br />
| bgcolor="#E0E0E0"| Sep 25 (Tues.)<br />
| bgcolor="#F0B0B0"| Daniel Ross <br />
| bgcolor="#BCE2FE"|[[NTS Fall 2012/Abstracts#September 25 | <font color="black"><em>tba</em></font>]]<br />
|- <br />
| bgcolor="#E0E0E0"| Oct 2 (Tues.)<br />
| bgcolor="#F0B0B0"| Lalit Jain <br />
| bgcolor="#BCE2FE"|[[NTS Fall 2012/Abstracts#October 2 | <font color="black"><em>tba</em></font>]]<br />
|- <br />
| bgcolor="#E0E0E0"| Oct 9 (Tues.)<br />
| bgcolor="#F0B0B0"| Daniel Ross <br />
| bgcolor="#BCE2FE"|[[NTS Fall 2012/Abstracts#October 9 | <font color="black"><em>tba</em></font>]]<br />
|- <br />
| bgcolor="#E0E0E0"| Oct 16 (Tues.)<br />
| bgcolor="#F0B0B0"| Evan Dummit <br />
| bgcolor="#BCE2FE"|[[NTS Fall 2012/Abstracts#October 16 | <font color="black"><em>tba</em></font>]]<br />
|- <br />
| bgcolor="#E0E0E0"| Oct 23 (Tues.)<br />
| bgcolor="#F0B0B0"| Ed Dewey<br />
| bgcolor="#BCE2FE"|[[NTS Fall 2012/Abstracts#October 23 | <font color="black"><em>tba</em></font>]]<br />
|- <br />
| bgcolor="#E0E0E0"| Oct 30 (Tues.)<br />
| bgcolor="#F0B0B0"| Yueke Hu <br />
| bgcolor="#BCE2FE"|[[NTS Fall 2012/Abstracts#October 30 | <font color="black"><em>tba</em></font>]]<br />
|- <br />
| bgcolor="#E0E0E0"| Nov 6 (Tues.)<br />
| bgcolor="#F0B0B0"| Evan Dummit<br />
| bgcolor="#BCE2FE"|[[NTS Fall 2012/Abstracts#November 6 | <font color="black"><em>tba</em></font>]]<br />
|- <br />
| bgcolor="#E0E0E0"| Nov 13 (Tues.)<br />
| bgcolor="#F0B0B0"| None!<br />
| bgcolor="#BCE2FE"|[[NTS Fall 2012/Abstracts#November 13 | <font color="black"><em>tba</em></font>]]<br />
|-<br />
| bgcolor="#E0E0E0"| Nov 20 (Tues.)<br />
| bgcolor="#F0B0B0"| (probably) None!<br />
| bgcolor="#BCE2FE"|[[NTS Fall 2012/Abstracts#November 20 | <font color="black"><em>tba</em></font>]]<br />
|-<br />
| bgcolor="#E0E0E0"| Nov 27 (Tues.)<br />
| bgcolor="#F0B0B0"| Megan Maguire <br />
| bgcolor="#BCE2FE"|[[NTS Fall 2012/Abstracts#November 27 | <font color="black"><em>tba</em></font>]]<br />
|- <br />
| bgcolor="#E0E0E0"| Dec 4 (Tues.)<br />
| bgcolor="#F0B0B0"| Peng Yu <br />
| bgcolor="#BCE2FE"|[[NTS Fall 2012/Abstracts#December 4 | <font color="black"><em>tba</em></font>]]<br />
|- <br />
| bgcolor="#E0E0E0"| Dec 11 (Tues.)<br />
| bgcolor="#F0B0B0"| Evan Dummit<br />
| bgcolor="#BCE2FE"|[[NTS Fall 2012/Abstracts#December 11 | <font color="black"><em>tba</em></font>]]<br />
|}<br />
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== Organizers ==<br />
<!--[http://www.math.wisc.edu/~rharron/ Robert Harron]<br />
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[http://www.math.wisc.edu/~klagsbru/ Zev Klagsbrun]<br />
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The Spring 2012 NTS Grad page can be found [[NTSGrad Spring 2012|here]].<br />
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Return to the [[NTS|Number Theory Seminar Page]]<br />
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Return to the [[Algebra|Algebra Group Page]]</div>Dummithttps://hilbert.math.wisc.edu/wiki/index.php?title=741&diff=47597412012-11-30T06:37:28Z<p>Dummit: /* HOMEWORK 10 (due Dec 4) */ typo fix</p>
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<div>'''Math 741'''<br />
<br />
Algebra<br />
<br />
Prof: [http://www.math.wisc.edu/~ellenber Jordan Ellenberg]<br />
<br />
Grader: Evan Dummit<br />
<br />
Ellenberg's office hours: Friday 3pm.<br />
<br />
Grader's office hours: Monday 4pm. Late homework may be given directly to the grader, along with either (i) the instructor's permission, or (ii) a polite request for mercy.<br />
<br />
This course, the first semester of the introductory graduate sequence in algebra, will cover the basic theory of groups, group actions, representation, linear and multilinear algebra, and the beginnings of ring theory.<br />
<br />
==SYLLABUS==<br />
<br />
In this space we will record the theorems and definitions we covered each week, which we can use as a list of notions you should be prepared to answer questions about on the Algebra qualifying exam. The material covered on the homework is also an excellent guide to the scope of the course. <br />
<br />
'''WEEK 1''': <br />
<br />
Definition of group. Associativity. Inverse. <br />
<br />
Examples of group: GL_n(R). GL_n(Z). Z/nZ. R. Z. R^*. The free group F_k on k generators. <br />
<br />
Homomorphisms. The homomorphisms from F_k to G are in bijection with G^k. Isomorphisms.<br />
<br />
'''WEEK 2''':<br />
<br />
The symmetric group (or permutation group) S_n on n letters. Cycle decomposition of a permutation. Order of a permutation. Thm: every element of a finite group has finite order. <br />
<br />
Subgroups. Left and right cosets. Lagrange's Theorem. Cyclic groups. The order of an element of a finite group is a divisor of the order of the group.<br />
<br />
The sign homomorphism S_n -> +-1.<br />
<br />
<br />
'''WEEK 3'''<br />
<br />
Normal subgroups. The quotient of a group by a normal subgroup. The first isomorphism theorem. Examples of S_n -> +-1 and S_4 -> S_3 with kernel V_4, the Klein 4-group.<br />
<br />
Centralizers and centers. Abelian groups. The center of SL_n(R) is either 1 or +-1.<br />
<br />
Groups with presentations. The infinite dihedral group <x,y | x^2 = 1, y^2 = 1>.<br />
<br />
'''WEEK 4'''<br />
<br />
More on groups with presentations.<br />
<br />
Second and third isomorphism theorems.<br />
<br />
Semidirect products.<br />
<br />
'''WEEK 5'''<br />
<br />
Group actions, orbits, and stabilizers.<br />
<br />
Orbit-stabilizer theorem.<br />
<br />
Cayley's theorem.<br />
<br />
Cauchy's theorem.<br />
<br />
'''WEEK 6'''<br />
<br />
Applications of orbit-stabilizer theorem (p-groups have nontrivial center, first Sylow theorem.)<br />
<br />
Classification of finite abelian groups and finitely generated abelian groups.<br />
<br />
Composition series and the Jordan-Holder theorem (which we state but don't prove.) <br />
<br />
The difference between knowing the composition factors and knowing the group (e.g. all p-groups of the same order have the same composition factors.)<br />
<br />
'''WEEK 7'''<br />
<br />
Simplicity of A_n.<br />
<br />
Nilpotent groups (main example: the Heisenberg group)<br />
<br />
Derived series and lower central series.<br />
<br />
Category theory 101: Definition of category and functor. Some examples. A group is a groupoid with one object.<br />
<br />
'''WEEK 8'''<br />
<br />
Introduction to representation theory.<br />
<br />
'''WEEK 10'''<br />
<br />
Ring theory 101: Rings, ring homomorphisms, ideals, isomorphism theorems. Examples: fields, Z, the Hamilton quaternions, matrix rings, rings of polynomials and formal power series, quadratic integer rings, group rings. Integral domains. Maximal and prime ideals. The nilradical.<br />
<br />
Module theory 101: Modules, module homomorphisms, submodules, isomorphism theorems. Noetherian modules and Zorn's lemma. Direct sums and direct products of arbitrary collections of modules.<br />
<br />
<br />
Below you will find a repository of homework problems. Note that some of these problems are taken from Lang's ''Algebra''.<br />
<br />
==HOMEWORK 1 (due Sep 20)==<br />
<br />
1. Suppose that H_1 and H_2 are subgroups of a group G. Prove that the intersection of H_1 and H_2 is a subgroup of G.<br />
<br />
2. Recall that S_3 (the symmetric group) is the group of permutations of the set {1..3}. List all the subgroups of S_3.<br />
<br />
3. We can define an equivalence relation on rational numbers by declaring two rational numbers to be equal whenever they differ by an integer. We denote the set of equivalence classes by Q/Z. The operation of addition makes Q/Z into a group.<br />
<br />
a) For each n, prove that Q/Z has a subgroup of order n.<br />
<br />
b) Prove that Q/Z is a ''divisible'' group: that is, if x is an element of Q/Z and n is an integer, there exists an element y of Q/Z such that ny = x. (Note that we write the operation in this group as addition rather than multiplication, which is why we write ny for the n-fold product of y with itself rather than y^n)<br />
<br />
c) Prove that Q/Z is not finitely generated. (Hint: prove that if x_1, .. x_d is a finite subset of Q/Z, the subgroup of Q/Z generated by x_1, ... x_d is finite.)<br />
<br />
4. Let F_2 be the free group on two generators x,y. Prove that there is a automorphism of F_2 which sends x to xyx and y to xy, and prove that this automorphism is unique.<br />
<br />
5. Let H be a subgroup of G, and let N_G(H), the ''normalizer'' of H in G, be the set of elements g in G satsifying g H g^{-1} = H. Prove that N_G(H) is a subgroup of G.<br />
<br />
6. Let T be the subgroup of GL_2(R) consisting of diagonal matrices. This is an example of a "Cartan subgroup," or "torus" (whence the notation T.) Describe the normalizer N(T) of T in GL_2(R). Prove that there is a homomorphism from N(T) to Z/2Z whose kernel is precisely T. <br />
<br />
7. Prove that the group of inner automorphisms of a group is normal in the full automorphism group Aut(G).<br />
<br />
8. Let H and H' be subgroups of a group G, and let x be an element of G. We denote by HxH' the set of elements of the form hxh', with h in H and h' in H', and we call HxH a ''double coset'' of the pair (H,H').<br />
<br />
a) Show that G decomposes as a disjoint union of double cosets of (H,H').<br />
<br />
b) Let G be GL_2(R) and let B be the subgroup of upper triangular matrices. Prove that G decomposes into exactly two double cosets of (B,B).<br />
<br />
c) Let G be the symmetric group S_n, and let H be the subgroup of permutations which fix 1. Describe the double coset decomposition of S_n into double cosets of (H,H).<br />
<br />
==HOMEWORK 2 (due Sep 27)==<br />
<br />
1. Prove that the symmetric group S_n is generated by transpositions. Hint: proceed by induction on n. Let H be the subgroup of S_n consisting of permutations fixing 1. Observe that H is isomorphic to S_{n-1}, so that H is generated by transpositions. Explain why it suffices to show that, for any g in S_n, the coset gH contains a transposition. Then explain why this is the case.<br />
<br />
2. Given that the symmetric group S_n is generated by transpositions, show that there are no nontrivial homomorphisms from S_n to Z/pZ for any odd prime p.<br />
<br />
3. Compute the centralizer of the triple flip (12)(34)(56) in S_6. (Note: this is the kind of problem that a computer algebra package like sage is very good at, and I encourage you to learn to use sage well enough to check your answer.)<br />
<br />
4. Warning: a normal subgroup of a normal subgroup need not be normal! Let H be the subgroup of S_4 generated by (12)(34). Then H is a subgroup of the Klein 4-group V_4, which we have already shown is normal in S_4. Show that H is normal in V_4, but H is not normal in S_4.<br />
<br />
5. The '''quaternion group''' Q is a non-abelian finite group of order 8. Its elements are 1, -1, i, -i, j, -j, k, -k; it satisfies g^2 = -1 for all g except +-1, and ijk = 1. (Note that the notation "-i" means "the product of i with -1" and -1 is understood to be central in Q.) Prove that every subgroup of Q is normal.<br />
<br />
6. (finishing example done in class) Let Gamma be the group F<x,y> / (x^2 = 1, y^2 = 1). (Remember, this means the quotient of the free group by the group generated by all conjugates of x^2 and y^2.) Prove that Gamma is infinite. Prove that Gamma has a subgroup R generated by xy isomorphic to Z, that R is normal, and that the quotient Gamma/R is isomorphic to Z/2Z. <br />
<br />
7. Suppose a group G has the property that every element in g satisfies g^2 = 1. Show that G must be abelian.<br />
<br />
==HOMEWORK 3 (due Oct 4)==<br />
<br />
1. Multiplication by -1 is an automorphism of Z/nZ. Let a be the homomorphism Z/2Z -> Aut(Z/nZ) which sends the nontrivial element of Z/2Z to multiplication by -1. Then we have a semidirect product of Z/pZ by Z/2Z as defined in class. This group of order 2n is called a ''dihedral group.'' and is denoted D_n (or sometimes D_{2n}).<br />
<br />
1a. Compute the center of D_n. (Note that the answer depends on n!)<br />
1b. Show that all involutions of D_n are conjugate to each other if and only if n is odd.<br />
<br />
2. (More on direct products that we didn't do in class.) Suppose that G is a semidirect product of N with H. Suppose furthermore that H is also normal in G. Prove that G is a direct product of N with H.<br />
<br />
3. Let Q be the quaternion group from last week's homework. Show that Q does not decompose as a semidirect product N \rtimes H, apart form the trivial cases N = {e} and N = Q.<br />
<br />
4. The ''affine linear group'' of degree n is the group of transformations from R^n to R^n of the form x -> Ax + b, where A is a matrix in GL_n(R) and b is a vector in R^n. Prove that the affine linear group is a semidirect product R^n \rtimes GL_n.<br />
<br />
5. The ''ordinary triangle group'' T(p,q,r) is the group with presentation <x,y | x^p = y^q = (xy)^r = 1>. This is a very interesting family of groups which arises naturally in many contexts in number theory and topology.<br />
<br />
6a. Prove that T(2,2,n) is isomorphic to the dihedral group D_n.<br />
6b. Prove that T(2,3,3) is finite and compute its order. Is it isomorphic to a finite group we've discussed in class?<br />
<br />
(It is a beautiful fact, but outside the scope of this course, that T(p,q,r) is finite if and only if 1/p + 1/q + 1/r > 1.)<br />
<br />
7. Let G be a subgroup of GL_n(R) which contains SL_n(R). Prove that G is a normal subgroup of GL_n(R).<br />
<br />
8. Let G be a group and let H_1 and H_2 be subgroups of G of finite index (that is, the number of cosets in G/H_i is finite, i = 1,2.) Prove that H_1 intersect H_2 also has finite index in G.<br />
<br />
9. Recall that the commutator [x,y] is x y x^{-1} y^{-1}. If G is a group, the commutator subgroup of G, denoted G' or [G,G], is the subgroup of G generated by all commutators [x,y] for x,y in G.<br />
<br />
9a. Show that G' is a normal subgroup of G.<br />
9b. Show that G/G' is an abelian group.<br />
9c. Show that if f: G -> A is a homomorphism, with A abelian, then G' is contained in ker f. Conclude that f factors through a homomorphism G/G' -> A.<br />
9d. Show that G' = G if and only if G has no nontrivial abelian quotient. In this case, we say G is ''perfect''.<br />
<br />
10. Under what conditions on (p,q,r) is the triangle group T(p,q,r) perfect?<br />
<br />
==HOMEWORK 4 (due Oct 16)==<br />
<br />
1. Let X be a set with a transitive action of a group G, which we think of as a homomorphism f: G -> Sym(X). Let H_x be the stabilizer of an element x of X.<br />
<br />
1a. If x' is another element of X, show that H_{x"} and H_x are conjugate subgroups. (Hint: this is not true without the hypothesis of transitivity, so make sure to use it!)<br />
<br />
1b. Show that the kernel of f is the intersection of all the conjugates of H_x in G. (This subgroup is called the ''normal core'' of H_x.)<br />
<br />
2. This week, a certified mathematical grown-up asked on MathOverflow what the index-3 subgroups of SL_2(Z/pZ) were. Let's do it ourselves for homework. Let p be a prime, and suppose H is an index-3 subgroup of SL_2(Z/pZ). I do not claim that the proof sketched below is necessarily the easiest, but it allows us to cover some ground that we won't get to do in lecture.<br />
<br />
2a. Using the coset action, show that the normal core of H is the kernel of a homomorphism f from SL_2(Z/pZ) to S_3.<br />
<br />
2b. A ''unipotent matrix'' is a matrix A such that (A-I)^n = 0 for some positive integer n. Prove that a unipotent element of SL_2(Z/pZ) has order dividing p.<br />
<br />
2c. You may use the fact that unipotent matrices generate SL_2(Z/pZ). Given this fact, show that f must be trivial when p > 3.<br />
<br />
2d. Using the above, show that SL_2(Z/pZ) has no index-3 subgroups when p > 3.<br />
<br />
2e. (extra credit) What can you say about the minimal index of a proper subgroup of SL_2(Z/pZ)?<br />
<br />
3. Let H be a subgroup of G of index 2. Prove that H is normal.<br />
<br />
4. We proved two theorems in class: Lagrange's theorem, which says that the order of a subgroup of G divides |G|, and Cauchy's theorem, which says that if p divides |G| then there exists a subgroup of G of order p. Cauchy's theorem is a kind of partial converse to Lagrange's theorem, but the full converse is false. Give an example of a finite group G and a divisor n of |G| such that you can prove there is no subgroup of G of order n.<br />
<br />
5. Let X be the set of lines in F_3^2 (here a line means a 1-dimensional linear subspace) so that |X| = 4. Let G be the projective linear group PGL_2(F_3), so that G acts on X via its action on F_3^2. This action can be thought of as a homomorphism from G to S_4. Show that this homomorphism is an ISOMORPHISM from PGL_2(F_3) to S_4.<br />
<br />
6. The group S_3 has only three conjugacy classes. Prove that a finite group with at most three conjugacy classes has order at most 6.<br />
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7. Let X be the set of ordered triples of elements of {1,..,n}, for some n >= 3. Then S_n acts on X. How many orbits are there? (Hint: the answer does not depend on n.)<br />
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8. Suppose that G and H are groups and f: G -> H is a homomorphism. Recall the definition of the abelianization G^ab from the previous problem set.<br />
<br />
8a. Show that the composition G -> H -> H^ab factors through a unique homomorphism G^ab -> H^ab, which we denote f^ab.<br />
<br />
8b. Show that if f: G -> H and g: H -> Q are homomorphisms, then f^ab o g^ab = (f o g)^ab.<br />
<br />
(For those reading MacLane, this constitutes a proof that abelianization is a '''functor''' from the category of groups to the category of abelian groups.)<br />
<br />
==HOMEWORK 5 (due Oct 23)==<br />
<br />
1. Show that there are only two different isomorphism classes of groups of order 14. (Hint: prove that such a group must be a semidirect product of a group of order 7 by a group of order 2.)<br />
<br />
2. (variant of Burnside's Lemma) Let G be a finite group and H a normal subgroup. Suppose G acts on a finite set X. <br />
<br />
2a. Show that if x and y lie in the same H-orbit, and g is an element of G, then gx and gy also lie in the same H-orbit. Thus, G/H has a natural action on the set of H-orbits X/H.<br />
<br />
2b. Show that the average number of fixed points in X of an element of the coset gH is the same as the number of fixed points of the element gH of G/H in its action on X/H.<br />
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3. A ''central extension'' of a group G by an abelian group A is a group E, together with a surjective homomorphism E -> G whose kernel is central in E and is isomorphic to A.<br />
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3a. Show that the quaternion group is a central extension of (Z/2Z)^2 by Z/2Z.<br />
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3b. Give an example of another nonabelian group which is a central extension of (Z/2Z)^2 by Z/2Z and which is not isomorphic to the quaternion group.<br />
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4. Prove that a central extension of an abelian group is nilpotent.<br />
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5. Give two different composition series for S_4 and show that they have the same composition factors.<br />
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6. Suppose that A is a finitely generated abelian group, and B is a subgroup of A.<br />
Prove that rank(A/B) = rank(A) - rank(B).<br />
ADDED: I asserted in class, but did not prove, that a subgroup of a finitely generated abelian group is again finitely generated. As one class member pointed out, without this fact it's not even clear why it makes sense to talk about rank(B)! You may assume it (though it would have been better for me to have included proving this fact as part of the problem....)<br />
<br />
7. For this problem, you may use the fact (which I should have asked on homework earlier) that if H_1 and H_2 are subgroups of G, then<br />
<br />
[G: H_1 intersect H_2] <= [G:H_1][G:H_2].<br />
<br />
Prove that if G is a simple group, d is an integer greater than 1, and S is a subgroup of index d in G, then |G| <= d^d.<br />
<br />
Conclude that, in particular, A_5 has no subgroup of index 3. On the other hand, exhibit a subgroup of A_5 of index 5.<br />
<br />
(Note: d^d is not a sharp bound; can you get a bound of d factorial instead?)<br />
<br />
<!--(This argument shows that a solvable subgroup of A_n has size at most about (n-1)!, but this is far from sharp; e.g. I found a citation to a 1967 paper of Dixon which shows that a solvable subgroup of A_n has size at most k^n where k is around 2.88.)--><br />
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8. Give an example of an abelian group A with a subgroup B such that B is not a direct summand of A.<br />
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9. Let A be the subgroup of Z^2 generated by the vector (1,0). Let B be the subgroup of Z^2 generated by the vector (3,3). Show that A and B are isomorphic to each other, but Z^2 / A is not isomorphic to Z^2 / B.<br />
<br />
==HOMEWORK 6 (due Oct 30)==<br />
<br />
1. In this problem, you will construct a functor F from the category of finite sets to the category of complex vector spaces. I'll tell you what it does on objects: if X is a finite set, then F(X) is the vector space spanned by a set of basis elements {e_x}_{x in X}, so that dim F(X) = |X|. Your job: complete the definition of F by describing the map from Mor_{Finite Sets}(X,Y) to Mor_{Vector Spaces}(F(X),F(Y)), and showing that your map respects composition.<br />
<br />
2. If (V, rho) is a representation of a group G, denote by V^G the space of vectors in V such that gv = v for all g in G; these are called _invariant_ vectors. We saw that the permutation representation of S_3 has a one-dimensional space of invariants. Prove that if X is a finite set with G-action, and V_X the corresponding permutation representation of G, then<br />
<br />
dim V_X^G = number of orbits of X.<br />
<br />
3. (Invariants are functorial) Let G be a group. Then there is a category called C[G]-Mod whose objects are complex representations of G (i.e. pairs (V,rho) where V is a complex vector space and rho is a homomorphism from G to GL(V)). The morphisms in this category are the ones described in class: Mor((V,rho),(W,psi)) is the set of linear maps f from V to W such that<br />
<br />
f(rho(g)(v)) = psi(g)(f(v))<br />
<br />
for all g in G and all v in V.<br />
<br />
Show that there is a functor H_0 from the category of representations of G to the category of vector spaces whose action on objects is given by<br />
<br />
H_0((V) = V^G.<br />
<br />
(In other words: your job again is to explain how to associate to a morphism from (V,rho) to (W,phi) a morphism from V^G to W^G, in a way that's compatible with composition.)<br />
<br />
<!--4. The group PSL_2(Z) has a presentation of the form [U,V|U^2 = V^3 = 1]. Thus, an n-dimensional representation of PSL_2(Z) is precisely a pair of n x n matrices (rho(U),rho(V)), such that rho(U)^2 = rho(V)^3 = 1. Prove that for every n > 1 there are uncountably many isomorphism classes of n-dimensional representations of PSL_2(Z). (Extra credit: show that there are uncountably many isomorphism classes of _irreducible_ representations.)<br />
<br />
An amazing theorem of Margulis (superrigidity) shows that SL_2(Z) is very special among arithmetic groups in this way. For instance, there are only _finitely_ many isomorphism classes of n-dimensional representations of PSL_d(Z) for any d > 2; those groups are "rigid."--><br />
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5. Let G be a finite group, let X be a set with G-action, and let chi be the character of the corresponding permutation representation. Show that, for each g in G, chi(g) is the number of elements of X fixed by g. (I am actually going to prove this in class.)<br />
<br />
6. Let V be the space of homogeneous degree-3 polynomials in variables x_1, x_2, x_3; then S_3 acts on V by permutation of the three variables. Describe the decomposition of V into irreducible representations of S_3.<br />
<br />
7a. Show that if chi is the character of a representation of a finite group G, then chi(g^{-1}) is the complex conjugate of chi.<br />
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7b. As a consequence, show that if g is conjugate to g^{-1} for all g in G, then chi(g) is real.<br />
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7c. The groups D_p and S_n have the property that g is conjugate to g^{-1} for _every_ g in G, so that every character is a real-valued function on G. Give an example of a group G where this is not the case, and give an example of a representation of G whose character takes non-real values.<br />
<br />
==HOMEWORK 7 (due Nov 6)==<br />
<br />
1. Let G be a finite group, let X be a finite set with G-action, and let chi be the character of the corresponding permutation representation. Show that, for each g in G, chi(g) is the number of elements of X fixed by g. (I am actually going to prove this in class.)<br />
<br />
2a. Show that if chi is the character of a finite-dimensional representation of a finite group G, then chi(g^{-1}) is the complex conjugate of chi(g) for all g.<br />
<br />
2b. As a consequence, show that if g is conjugate to g^{-1} for all g in G, then chi(g) is real.<br />
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2c. The groups D_p and S_n have the property that g is conjugate to g^{-1} for _every_ g in G, so that every character is a real-valued function on G. Give an example of a group G where this is not the case, and give an example of a representation of G whose character takes non-real values.<br />
<br />
3. Let V and W be complex vector spaces. Let f be a linear transformation of V, and g a linear transformation of W. <br />
<br />
3a. Show that there is a unique linear transformation F satisfying<br />
<br />
F(v tensor w) = f(v) tensor g(w)<br />
<br />
for all v in V and all w in W. We denote this transformation by f tensor g.<br />
<br />
3b. Suppose V and W are finite dimensional. Show that Trace(f tensor g) = Trace(f) Trace(g). What does this say when f and g are both the identity transformation?<br />
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4. Let V be a vector space. We define Sym^2 V to be the quotient of V tensor V by the subspace generated by all elements of the form<br />
<br />
v tensor w - w tensor v<br />
<br />
for v,w in V.<br />
<br />
Suppose dim V = n. What is dim Sym^2 V?<br />
<br />
5. Show that Z/pZ tensor Z/qZ is zero when p and q are distinct primes.<br />
<br />
==HOMEWORK 8 (due Nov 13)==<br />
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1. (Schur's lemma can be false over non-algebraically closed fields) Let G be Z/3Z and let g be a generator of G. Then G acts by cyclic permutations on the set {1,2,3}; let P be the corresponding permutation representation on the REAL (not complex!) vector space generated by e_1,e_2,e_3.<br />
<br />
1a. Show that P breaks up as the direct sum of the trivial representation and a 2-dimensional representation V which is irreducible. (Remember, we are not working over the complex numbers, so you cannot use the criterion for irreducibility in terms of the character that we worked out in class for the complex case; you need to prove directly that V has no nontrivial proper subrepresentation.)<br />
<br />
1b. Show that the map from V to V induced by g is G-equivariant, but is neither zero nor a scalar multiplication.<br />
<br />
2. If V_1 and V_2 are representations (not necessarily irreducible!) of G, show that <chi_V_1, chi V_2> is a non-negative integer.<br />
<br />
3. When a formula always yields a non-negative integer, it is often because it is secretly a formula for either the cardinality of some finite set or the dimension of some finite-dimensional vector space. In this case, show that<br />
<br />
<chi_{V_1}, chi_{V_2}> = dim_C Hom_G(V_1,V_2)<br />
<br />
where we recall that Hom_G(V_1,G_2) refers to the space of G-equivariant homomorphisms from V_1 to V_2.<br />
<br />
4. If A is an abelian group, show that every irreducible representation of A is 1-dimensional. (Hint: show that there is a vector which is an eigenvector for every element of A!)<br />
<br />
5. Let G be the group of order 20 which is the semidirect product of A by H, where A is Z/5Z and H is (Z/5Z)^* with its natural action on A. We will work out all the irreducible representations of A. <br />
<br />
5a. First of all, consider the 5-dimensional permutation representation of G acting by left multiplication on the cosets of H. Show that this representation decomposes as a direct sum of the trivial representation and an irreducible 4-dimensional representation. (Hint: it is probably easier to use the character of this representation than to prove irreducibility directly.) <br />
<br />
5b. Now show that there are 4 1-dimensional representations of G which factor through the quotient G/A (i.e. such that rho(A) is the identity.)<br />
<br />
5c. Using the decomposition of the regular representation, show that the irreducible representations of G you have constructed are the only ones.<br />
<br />
==HOMEWORK 9 (due Nov 20)==<br />
<br />
0. Find all ring homomorphisms from Z to Z/30Z. [Warning: there are 8 of them, not 30.]<br />
<br />
<br />
1. If R is a ring (not necessarily with 1) such that a^2 = a for every a in R, R is called a Boolean ring.<br />
<br />
1a. Show that any nonzero Boolean ring has characteristic 2. (In other words, show that a+a=0 for all a in R.)<br />
<br />
1b. Show that every Boolean ring is commutative.<br />
<br />
<br />
2. Let phi : R -> S be a homomorphism of commutative rings with 1.<br />
<br />
2a. If I is an ideal of S, show that phi^(-1) (I) is an ideal of R.<br />
<br />
2b. If P is a prime ideal of S, show that phi^(-1) (P) is either R or a prime ideal of R.<br />
<br />
2c. If M is a maximal ideal of S and phi is surjective, show that phi^(-1) (M) is a maximal ideal of R.<br />
<br />
2d. Find a homomorphism phi : R -> S and a maximal ideal M of S such that phi^(-1) (M) is not a maximal ideal of R.<br />
<br />
<br />
3. Let Nil(A) denote the set of nilpotent elements of a ring A, and let R be a commutative ring. In class it was observed that Nil(R) is an ideal of R.<br />
<br />
3a. Find (a generator of) Nil(Z/720Z).<br />
<br />
3b. Show that Nil(R/Nil(R)) = 0.<br />
<br />
3c. If x is in Nil(R) and u is a unit, show that u+x is a unit.<br />
<br />
3d. Show that Nil(S), where S is the 2x2 matrices with real coefficients, is not an ideal of S.<br />
<br />
<br />
4a. If M is an R-module and I is an ideal of R such that im = 0 for all i in I and m in M, show that M carries the structure of an R/I-module.<br />
<br />
4b. If S is a commutative ring with 1 and J is a maximal ideal of S, show that J/J^2 is a vector space over S/J.<br />
<br />
<br />
5. A "divisible element" x of a Z-module is one such that for each positive integer k, there exists a y such that ky=x. Show that a free Z-module cannot contain a nonzero divisible element.<br />
<br />
<br />
6. Prove that the direct sum of any collection of free R-modules is also a free R-module.<br />
<br />
<br />
7. In this problem you will prove that the module M given by the direct product of copies of Z, indexed by the positive integers, is not a free Z-module. Therefore, assume that M is a free Z-module with basis B, and let N be the direct sum of copies of Z, as a submodule of M.<br />
<br />
7a. Show there is a countable subset B1 of B such that N is contained in the submodule N1 generated by B1. [Hint: N is countable.]<br />
<br />
7b. Show that M/N1 is a free Z-module.<br />
<br />
7c. Let S = {(b1, b2, b3, ...) : b_i = i! or -i!}. Show that there is some s in S which is not in N1. [Hint: S is uncountable.]<br />
<br />
7d. For any s in S but not N1, show that for each positive integer k, there exists some m in M with s-km in N1. Conclude that s+N1 is a divisible element of M/N1.<br />
<br />
7e. Use problem 5 to obtain a contradiction.<br />
<br />
<br />
8. In class I said "rank" is not always well-defined for R-modules if R is not commutative. Here is an example demonstrating this assertion. Let M be the countably infinite direct product of copies of Z as in exercise 7, and let R be the endomorphism ring of M. Define phi_1(a1,a2,a3,a4,...) = (a1,a3,a5,...) and phi_2(a1,a2,a3,a4,...) = (a2,a4,a6,...).<br />
<br />
8a. Show that {phi_1, phi_2} is a free basis for R, as a left R-module. [Hint: let psi_1(a1,a2,a3,...) = (a1,0,a2,0,...) and psi_2(a1,a2,a3,...) = (0,a1,0,a2,...). Compute the products of the phi_i and psi_j in either order, and use the result to show that the phi_i are a basis.]<br />
<br />
8b. Show that R is isomorphic to R^n for every positive integer n.<br />
<br />
<br />
==HOMEWORK 10 (due Dec 4)==<br />
<br />
1. '''How algebraists do calculus'''. The ''ring of dual numbers'' is the ring R = C[e]/(e^2). We think of e as an infinitesimal number; that is, something so small that its square is 0. This algebraic notion of infinitesimal is the basis for the algebraic notion of differentiation, which works as follows. Let f be a polynomial in R[x]. Show that there is a unique polynomial g in C[x] such that<br />
<br />
f(x+e) - f(x) = eg(x)<br />
<br />
and that in fact g is the derivative of f. In other words, the usual definition of "derivative" works just fine in this context, without any use of the notion of limit!<br />
<br />
2. Write down all the ideals of C[e]/e^2. (There are three.)<br />
<br />
3. Show that any right ideal of the ring M_2(Q) of 2x2 matrices is an ideal of the form I_V described in class for some subspace V of Q^2; that is, it consists precisely of those linear transformations whose image is contained in V.<br />
<br />
<br />
4. An ''idempotent'' element of a ring is an element e satisfying e^2 = e.<br />
<br />
4a. If e is an idempotent, show that 1-e is also an idempotent.<br />
<br />
4b. Suppose R has a unit (we are actually always supposing this, but I just want to remind you of it for this problem) and suppose e is a ''central'' idempotent, i.e. an idempotent which commutes with all elements of R. Show that eR and (1-e)R are both two-sided ideals. Moreover, show that eR and (1-e)R are also subrings of R (with the warning that the identity in eR is not 1, which is not contained in eR, but rather e.)<br />
<br />
4c. Show that the map f_e: R -> eR defined by f_e(r) = er is a ring homomorphism. Finally show that the ring homomorphism<br />
<br />
f_{e} x f_{1-e}: R -> eR x (1-e)R <br />
<br />
is actually an isomorphism of rings. In other words, the presence of a central idempotent induces a decomposition a ring as a direct product.<br />
<br />
<br />
5. The ''center'' of a ring R is the subring of elements commuting with every element of R.<br />
<br />
5a. Show that the center of M_n(Q) is the ring of scalar matrices.<br />
<br />
5b. Show that the center of the group ring C[G] is the set of elements sum f(g) g such that f(g) is a class function.<br />
<br />
<br />
6. '''How algebraists do differential equations.''' We define the ''Weyl algebra'' W of differential operators in one variable as follows. Multiplication by x is a linear transformation on the complex vector space V of all smooth functions on C. Differentiation, which we denote d, is also a linear transformation on V. So we let W be the subring of End(V) generated by x and d. We note that x and d satisfy the relation xd = dx - 1; in particular, the Weyl algebra is not commutative.<br />
<br />
6a. Describe the submodule of V which is annihilated by the element d-x in W.<br />
<br />
6b. Show that, for any polynomial f, fd -df = f'. (This is another way algebraists do calculus!) Use this fact to show that the center of the Weyl algebra is just the field of constants. (Remark: you will use in the course of this proof the fact that a polynomial whose derivative is 0 is constant -- this would not be true if we were working in positive characteristic, as the example of x^p considered as a polynomial over F_p demonstrates.)<br />
<br />
<br />
7. If M is a left module for a ring R, we denote by Ann(M) the set of elements r in R such that rm = 0 for all m in M. Show that Ann(M) is a two-sided ideal. <br />
<br />
<br />
8. Let M be a left module for R, and let m be an element of M. We denote by Ann(m) the set of elements r in R such that rm = 0.<br />
<br />
8a. Show that Ann(m) is a left ideal, and show that the submodule Rv of M generated by v is isomorphic to the left module R/Ann(m).<br />
<br />
8c. Let V,W be as in question 6. Show that Ann(e^x) is the left ideal W(d-1).</div>Dummithttps://hilbert.math.wisc.edu/wiki/index.php?title=741&diff=47587412012-11-29T23:57:03Z<p>Dummit: /* HOMEWORK 10 (due Dec 4) */ fixed typo, formatting</p>
<hr />
<div>'''Math 741'''<br />
<br />
Algebra<br />
<br />
Prof: [http://www.math.wisc.edu/~ellenber Jordan Ellenberg]<br />
<br />
Grader: Evan Dummit<br />
<br />
Ellenberg's office hours: Friday 3pm.<br />
<br />
Grader's office hours: Monday 4pm. Late homework may be given directly to the grader, along with either (i) the instructor's permission, or (ii) a polite request for mercy.<br />
<br />
This course, the first semester of the introductory graduate sequence in algebra, will cover the basic theory of groups, group actions, representation, linear and multilinear algebra, and the beginnings of ring theory.<br />
<br />
==SYLLABUS==<br />
<br />
In this space we will record the theorems and definitions we covered each week, which we can use as a list of notions you should be prepared to answer questions about on the Algebra qualifying exam. The material covered on the homework is also an excellent guide to the scope of the course. <br />
<br />
'''WEEK 1''': <br />
<br />
Definition of group. Associativity. Inverse. <br />
<br />
Examples of group: GL_n(R). GL_n(Z). Z/nZ. R. Z. R^*. The free group F_k on k generators. <br />
<br />
Homomorphisms. The homomorphisms from F_k to G are in bijection with G^k. Isomorphisms.<br />
<br />
'''WEEK 2''':<br />
<br />
The symmetric group (or permutation group) S_n on n letters. Cycle decomposition of a permutation. Order of a permutation. Thm: every element of a finite group has finite order. <br />
<br />
Subgroups. Left and right cosets. Lagrange's Theorem. Cyclic groups. The order of an element of a finite group is a divisor of the order of the group.<br />
<br />
The sign homomorphism S_n -> +-1.<br />
<br />
<br />
'''WEEK 3'''<br />
<br />
Normal subgroups. The quotient of a group by a normal subgroup. The first isomorphism theorem. Examples of S_n -> +-1 and S_4 -> S_3 with kernel V_4, the Klein 4-group.<br />
<br />
Centralizers and centers. Abelian groups. The center of SL_n(R) is either 1 or +-1.<br />
<br />
Groups with presentations. The infinite dihedral group <x,y | x^2 = 1, y^2 = 1>.<br />
<br />
'''WEEK 4'''<br />
<br />
More on groups with presentations.<br />
<br />
Second and third isomorphism theorems.<br />
<br />
Semidirect products.<br />
<br />
'''WEEK 5'''<br />
<br />
Group actions, orbits, and stabilizers.<br />
<br />
Orbit-stabilizer theorem.<br />
<br />
Cayley's theorem.<br />
<br />
Cauchy's theorem.<br />
<br />
'''WEEK 6'''<br />
<br />
Applications of orbit-stabilizer theorem (p-groups have nontrivial center, first Sylow theorem.)<br />
<br />
Classification of finite abelian groups and finitely generated abelian groups.<br />
<br />
Composition series and the Jordan-Holder theorem (which we state but don't prove.) <br />
<br />
The difference between knowing the composition factors and knowing the group (e.g. all p-groups of the same order have the same composition factors.)<br />
<br />
'''WEEK 7'''<br />
<br />
Simplicity of A_n.<br />
<br />
Nilpotent groups (main example: the Heisenberg group)<br />
<br />
Derived series and lower central series.<br />
<br />
Category theory 101: Definition of category and functor. Some examples. A group is a groupoid with one object.<br />
<br />
'''WEEK 8'''<br />
<br />
Introduction to representation theory.<br />
<br />
'''WEEK 10'''<br />
<br />
Ring theory 101: Rings, ring homomorphisms, ideals, isomorphism theorems. Examples: fields, Z, the Hamilton quaternions, matrix rings, rings of polynomials and formal power series, quadratic integer rings, group rings. Integral domains. Maximal and prime ideals. The nilradical.<br />
<br />
Module theory 101: Modules, module homomorphisms, submodules, isomorphism theorems. Noetherian modules and Zorn's lemma. Direct sums and direct products of arbitrary collections of modules.<br />
<br />
<br />
Below you will find a repository of homework problems. Note that some of these problems are taken from Lang's ''Algebra''.<br />
<br />
==HOMEWORK 1 (due Sep 20)==<br />
<br />
1. Suppose that H_1 and H_2 are subgroups of a group G. Prove that the intersection of H_1 and H_2 is a subgroup of G.<br />
<br />
2. Recall that S_3 (the symmetric group) is the group of permutations of the set {1..3}. List all the subgroups of S_3.<br />
<br />
3. We can define an equivalence relation on rational numbers by declaring two rational numbers to be equal whenever they differ by an integer. We denote the set of equivalence classes by Q/Z. The operation of addition makes Q/Z into a group.<br />
<br />
a) For each n, prove that Q/Z has a subgroup of order n.<br />
<br />
b) Prove that Q/Z is a ''divisible'' group: that is, if x is an element of Q/Z and n is an integer, there exists an element y of Q/Z such that ny = x. (Note that we write the operation in this group as addition rather than multiplication, which is why we write ny for the n-fold product of y with itself rather than y^n)<br />
<br />
c) Prove that Q/Z is not finitely generated. (Hint: prove that if x_1, .. x_d is a finite subset of Q/Z, the subgroup of Q/Z generated by x_1, ... x_d is finite.)<br />
<br />
4. Let F_2 be the free group on two generators x,y. Prove that there is a automorphism of F_2 which sends x to xyx and y to xy, and prove that this automorphism is unique.<br />
<br />
5. Let H be a subgroup of G, and let N_G(H), the ''normalizer'' of H in G, be the set of elements g in G satsifying g H g^{-1} = H. Prove that N_G(H) is a subgroup of G.<br />
<br />
6. Let T be the subgroup of GL_2(R) consisting of diagonal matrices. This is an example of a "Cartan subgroup," or "torus" (whence the notation T.) Describe the normalizer N(T) of T in GL_2(R). Prove that there is a homomorphism from N(T) to Z/2Z whose kernel is precisely T. <br />
<br />
7. Prove that the group of inner automorphisms of a group is normal in the full automorphism group Aut(G).<br />
<br />
8. Let H and H' be subgroups of a group G, and let x be an element of G. We denote by HxH' the set of elements of the form hxh', with h in H and h' in H', and we call HxH a ''double coset'' of the pair (H,H').<br />
<br />
a) Show that G decomposes as a disjoint union of double cosets of (H,H').<br />
<br />
b) Let G be GL_2(R) and let B be the subgroup of upper triangular matrices. Prove that G decomposes into exactly two double cosets of (B,B).<br />
<br />
c) Let G be the symmetric group S_n, and let H be the subgroup of permutations which fix 1. Describe the double coset decomposition of S_n into double cosets of (H,H).<br />
<br />
==HOMEWORK 2 (due Sep 27)==<br />
<br />
1. Prove that the symmetric group S_n is generated by transpositions. Hint: proceed by induction on n. Let H be the subgroup of S_n consisting of permutations fixing 1. Observe that H is isomorphic to S_{n-1}, so that H is generated by transpositions. Explain why it suffices to show that, for any g in S_n, the coset gH contains a transposition. Then explain why this is the case.<br />
<br />
2. Given that the symmetric group S_n is generated by transpositions, show that there are no nontrivial homomorphisms from S_n to Z/pZ for any odd prime p.<br />
<br />
3. Compute the centralizer of the triple flip (12)(34)(56) in S_6. (Note: this is the kind of problem that a computer algebra package like sage is very good at, and I encourage you to learn to use sage well enough to check your answer.)<br />
<br />
4. Warning: a normal subgroup of a normal subgroup need not be normal! Let H be the subgroup of S_4 generated by (12)(34). Then H is a subgroup of the Klein 4-group V_4, which we have already shown is normal in S_4. Show that H is normal in V_4, but H is not normal in S_4.<br />
<br />
5. The '''quaternion group''' Q is a non-abelian finite group of order 8. Its elements are 1, -1, i, -i, j, -j, k, -k; it satisfies g^2 = -1 for all g except +-1, and ijk = 1. (Note that the notation "-i" means "the product of i with -1" and -1 is understood to be central in Q.) Prove that every subgroup of Q is normal.<br />
<br />
6. (finishing example done in class) Let Gamma be the group F<x,y> / (x^2 = 1, y^2 = 1). (Remember, this means the quotient of the free group by the group generated by all conjugates of x^2 and y^2.) Prove that Gamma is infinite. Prove that Gamma has a subgroup R generated by xy isomorphic to Z, that R is normal, and that the quotient Gamma/R is isomorphic to Z/2Z. <br />
<br />
7. Suppose a group G has the property that every element in g satisfies g^2 = 1. Show that G must be abelian.<br />
<br />
==HOMEWORK 3 (due Oct 4)==<br />
<br />
1. Multiplication by -1 is an automorphism of Z/nZ. Let a be the homomorphism Z/2Z -> Aut(Z/nZ) which sends the nontrivial element of Z/2Z to multiplication by -1. Then we have a semidirect product of Z/pZ by Z/2Z as defined in class. This group of order 2n is called a ''dihedral group.'' and is denoted D_n (or sometimes D_{2n}).<br />
<br />
1a. Compute the center of D_n. (Note that the answer depends on n!)<br />
1b. Show that all involutions of D_n are conjugate to each other if and only if n is odd.<br />
<br />
2. (More on direct products that we didn't do in class.) Suppose that G is a semidirect product of N with H. Suppose furthermore that H is also normal in G. Prove that G is a direct product of N with H.<br />
<br />
3. Let Q be the quaternion group from last week's homework. Show that Q does not decompose as a semidirect product N \rtimes H, apart form the trivial cases N = {e} and N = Q.<br />
<br />
4. The ''affine linear group'' of degree n is the group of transformations from R^n to R^n of the form x -> Ax + b, where A is a matrix in GL_n(R) and b is a vector in R^n. Prove that the affine linear group is a semidirect product R^n \rtimes GL_n.<br />
<br />
5. The ''ordinary triangle group'' T(p,q,r) is the group with presentation <x,y | x^p = y^q = (xy)^r = 1>. This is a very interesting family of groups which arises naturally in many contexts in number theory and topology.<br />
<br />
6a. Prove that T(2,2,n) is isomorphic to the dihedral group D_n.<br />
6b. Prove that T(2,3,3) is finite and compute its order. Is it isomorphic to a finite group we've discussed in class?<br />
<br />
(It is a beautiful fact, but outside the scope of this course, that T(p,q,r) is finite if and only if 1/p + 1/q + 1/r > 1.)<br />
<br />
7. Let G be a subgroup of GL_n(R) which contains SL_n(R). Prove that G is a normal subgroup of GL_n(R).<br />
<br />
8. Let G be a group and let H_1 and H_2 be subgroups of G of finite index (that is, the number of cosets in G/H_i is finite, i = 1,2.) Prove that H_1 intersect H_2 also has finite index in G.<br />
<br />
9. Recall that the commutator [x,y] is x y x^{-1} y^{-1}. If G is a group, the commutator subgroup of G, denoted G' or [G,G], is the subgroup of G generated by all commutators [x,y] for x,y in G.<br />
<br />
9a. Show that G' is a normal subgroup of G.<br />
9b. Show that G/G' is an abelian group.<br />
9c. Show that if f: G -> A is a homomorphism, with A abelian, then G' is contained in ker f. Conclude that f factors through a homomorphism G/G' -> A.<br />
9d. Show that G' = G if and only if G has no nontrivial abelian quotient. In this case, we say G is ''perfect''.<br />
<br />
10. Under what conditions on (p,q,r) is the triangle group T(p,q,r) perfect?<br />
<br />
==HOMEWORK 4 (due Oct 16)==<br />
<br />
1. Let X be a set with a transitive action of a group G, which we think of as a homomorphism f: G -> Sym(X). Let H_x be the stabilizer of an element x of X.<br />
<br />
1a. If x' is another element of X, show that H_{x"} and H_x are conjugate subgroups. (Hint: this is not true without the hypothesis of transitivity, so make sure to use it!)<br />
<br />
1b. Show that the kernel of f is the intersection of all the conjugates of H_x in G. (This subgroup is called the ''normal core'' of H_x.)<br />
<br />
2. This week, a certified mathematical grown-up asked on MathOverflow what the index-3 subgroups of SL_2(Z/pZ) were. Let's do it ourselves for homework. Let p be a prime, and suppose H is an index-3 subgroup of SL_2(Z/pZ). I do not claim that the proof sketched below is necessarily the easiest, but it allows us to cover some ground that we won't get to do in lecture.<br />
<br />
2a. Using the coset action, show that the normal core of H is the kernel of a homomorphism f from SL_2(Z/pZ) to S_3.<br />
<br />
2b. A ''unipotent matrix'' is a matrix A such that (A-I)^n = 0 for some positive integer n. Prove that a unipotent element of SL_2(Z/pZ) has order dividing p.<br />
<br />
2c. You may use the fact that unipotent matrices generate SL_2(Z/pZ). Given this fact, show that f must be trivial when p > 3.<br />
<br />
2d. Using the above, show that SL_2(Z/pZ) has no index-3 subgroups when p > 3.<br />
<br />
2e. (extra credit) What can you say about the minimal index of a proper subgroup of SL_2(Z/pZ)?<br />
<br />
3. Let H be a subgroup of G of index 2. Prove that H is normal.<br />
<br />
4. We proved two theorems in class: Lagrange's theorem, which says that the order of a subgroup of G divides |G|, and Cauchy's theorem, which says that if p divides |G| then there exists a subgroup of G of order p. Cauchy's theorem is a kind of partial converse to Lagrange's theorem, but the full converse is false. Give an example of a finite group G and a divisor n of |G| such that you can prove there is no subgroup of G of order n.<br />
<br />
5. Let X be the set of lines in F_3^2 (here a line means a 1-dimensional linear subspace) so that |X| = 4. Let G be the projective linear group PGL_2(F_3), so that G acts on X via its action on F_3^2. This action can be thought of as a homomorphism from G to S_4. Show that this homomorphism is an ISOMORPHISM from PGL_2(F_3) to S_4.<br />
<br />
6. The group S_3 has only three conjugacy classes. Prove that a finite group with at most three conjugacy classes has order at most 6.<br />
<br />
7. Let X be the set of ordered triples of elements of {1,..,n}, for some n >= 3. Then S_n acts on X. How many orbits are there? (Hint: the answer does not depend on n.)<br />
<br />
8. Suppose that G and H are groups and f: G -> H is a homomorphism. Recall the definition of the abelianization G^ab from the previous problem set.<br />
<br />
8a. Show that the composition G -> H -> H^ab factors through a unique homomorphism G^ab -> H^ab, which we denote f^ab.<br />
<br />
8b. Show that if f: G -> H and g: H -> Q are homomorphisms, then f^ab o g^ab = (f o g)^ab.<br />
<br />
(For those reading MacLane, this constitutes a proof that abelianization is a '''functor''' from the category of groups to the category of abelian groups.)<br />
<br />
==HOMEWORK 5 (due Oct 23)==<br />
<br />
1. Show that there are only two different isomorphism classes of groups of order 14. (Hint: prove that such a group must be a semidirect product of a group of order 7 by a group of order 2.)<br />
<br />
2. (variant of Burnside's Lemma) Let G be a finite group and H a normal subgroup. Suppose G acts on a finite set X. <br />
<br />
2a. Show that if x and y lie in the same H-orbit, and g is an element of G, then gx and gy also lie in the same H-orbit. Thus, G/H has a natural action on the set of H-orbits X/H.<br />
<br />
2b. Show that the average number of fixed points in X of an element of the coset gH is the same as the number of fixed points of the element gH of G/H in its action on X/H.<br />
<br />
3. A ''central extension'' of a group G by an abelian group A is a group E, together with a surjective homomorphism E -> G whose kernel is central in E and is isomorphic to A.<br />
<br />
3a. Show that the quaternion group is a central extension of (Z/2Z)^2 by Z/2Z.<br />
<br />
3b. Give an example of another nonabelian group which is a central extension of (Z/2Z)^2 by Z/2Z and which is not isomorphic to the quaternion group.<br />
<br />
4. Prove that a central extension of an abelian group is nilpotent.<br />
<br />
5. Give two different composition series for S_4 and show that they have the same composition factors.<br />
<br />
6. Suppose that A is a finitely generated abelian group, and B is a subgroup of A.<br />
Prove that rank(A/B) = rank(A) - rank(B).<br />
ADDED: I asserted in class, but did not prove, that a subgroup of a finitely generated abelian group is again finitely generated. As one class member pointed out, without this fact it's not even clear why it makes sense to talk about rank(B)! You may assume it (though it would have been better for me to have included proving this fact as part of the problem....)<br />
<br />
7. For this problem, you may use the fact (which I should have asked on homework earlier) that if H_1 and H_2 are subgroups of G, then<br />
<br />
[G: H_1 intersect H_2] <= [G:H_1][G:H_2].<br />
<br />
Prove that if G is a simple group, d is an integer greater than 1, and S is a subgroup of index d in G, then |G| <= d^d.<br />
<br />
Conclude that, in particular, A_5 has no subgroup of index 3. On the other hand, exhibit a subgroup of A_5 of index 5.<br />
<br />
(Note: d^d is not a sharp bound; can you get a bound of d factorial instead?)<br />
<br />
<!--(This argument shows that a solvable subgroup of A_n has size at most about (n-1)!, but this is far from sharp; e.g. I found a citation to a 1967 paper of Dixon which shows that a solvable subgroup of A_n has size at most k^n where k is around 2.88.)--><br />
<br />
8. Give an example of an abelian group A with a subgroup B such that B is not a direct summand of A.<br />
<br />
9. Let A be the subgroup of Z^2 generated by the vector (1,0). Let B be the subgroup of Z^2 generated by the vector (3,3). Show that A and B are isomorphic to each other, but Z^2 / A is not isomorphic to Z^2 / B.<br />
<br />
==HOMEWORK 6 (due Oct 30)==<br />
<br />
1. In this problem, you will construct a functor F from the category of finite sets to the category of complex vector spaces. I'll tell you what it does on objects: if X is a finite set, then F(X) is the vector space spanned by a set of basis elements {e_x}_{x in X}, so that dim F(X) = |X|. Your job: complete the definition of F by describing the map from Mor_{Finite Sets}(X,Y) to Mor_{Vector Spaces}(F(X),F(Y)), and showing that your map respects composition.<br />
<br />
2. If (V, rho) is a representation of a group G, denote by V^G the space of vectors in V such that gv = v for all g in G; these are called _invariant_ vectors. We saw that the permutation representation of S_3 has a one-dimensional space of invariants. Prove that if X is a finite set with G-action, and V_X the corresponding permutation representation of G, then<br />
<br />
dim V_X^G = number of orbits of X.<br />
<br />
3. (Invariants are functorial) Let G be a group. Then there is a category called C[G]-Mod whose objects are complex representations of G (i.e. pairs (V,rho) where V is a complex vector space and rho is a homomorphism from G to GL(V)). The morphisms in this category are the ones described in class: Mor((V,rho),(W,psi)) is the set of linear maps f from V to W such that<br />
<br />
f(rho(g)(v)) = psi(g)(f(v))<br />
<br />
for all g in G and all v in V.<br />
<br />
Show that there is a functor H_0 from the category of representations of G to the category of vector spaces whose action on objects is given by<br />
<br />
H_0((V) = V^G.<br />
<br />
(In other words: your job again is to explain how to associate to a morphism from (V,rho) to (W,phi) a morphism from V^G to W^G, in a way that's compatible with composition.)<br />
<br />
<!--4. The group PSL_2(Z) has a presentation of the form [U,V|U^2 = V^3 = 1]. Thus, an n-dimensional representation of PSL_2(Z) is precisely a pair of n x n matrices (rho(U),rho(V)), such that rho(U)^2 = rho(V)^3 = 1. Prove that for every n > 1 there are uncountably many isomorphism classes of n-dimensional representations of PSL_2(Z). (Extra credit: show that there are uncountably many isomorphism classes of _irreducible_ representations.)<br />
<br />
An amazing theorem of Margulis (superrigidity) shows that SL_2(Z) is very special among arithmetic groups in this way. For instance, there are only _finitely_ many isomorphism classes of n-dimensional representations of PSL_d(Z) for any d > 2; those groups are "rigid."--><br />
<br />
5. Let G be a finite group, let X be a set with G-action, and let chi be the character of the corresponding permutation representation. Show that, for each g in G, chi(g) is the number of elements of X fixed by g. (I am actually going to prove this in class.)<br />
<br />
6. Let V be the space of homogeneous degree-3 polynomials in variables x_1, x_2, x_3; then S_3 acts on V by permutation of the three variables. Describe the decomposition of V into irreducible representations of S_3.<br />
<br />
7a. Show that if chi is the character of a representation of a finite group G, then chi(g^{-1}) is the complex conjugate of chi.<br />
<br />
7b. As a consequence, show that if g is conjugate to g^{-1} for all g in G, then chi(g) is real.<br />
<br />
7c. The groups D_p and S_n have the property that g is conjugate to g^{-1} for _every_ g in G, so that every character is a real-valued function on G. Give an example of a group G where this is not the case, and give an example of a representation of G whose character takes non-real values.<br />
<br />
==HOMEWORK 7 (due Nov 6)==<br />
<br />
1. Let G be a finite group, let X be a finite set with G-action, and let chi be the character of the corresponding permutation representation. Show that, for each g in G, chi(g) is the number of elements of X fixed by g. (I am actually going to prove this in class.)<br />
<br />
2a. Show that if chi is the character of a finite-dimensional representation of a finite group G, then chi(g^{-1}) is the complex conjugate of chi(g) for all g.<br />
<br />
2b. As a consequence, show that if g is conjugate to g^{-1} for all g in G, then chi(g) is real.<br />
<br />
2c. The groups D_p and S_n have the property that g is conjugate to g^{-1} for _every_ g in G, so that every character is a real-valued function on G. Give an example of a group G where this is not the case, and give an example of a representation of G whose character takes non-real values.<br />
<br />
3. Let V and W be complex vector spaces. Let f be a linear transformation of V, and g a linear transformation of W. <br />
<br />
3a. Show that there is a unique linear transformation F satisfying<br />
<br />
F(v tensor w) = f(v) tensor g(w)<br />
<br />
for all v in V and all w in W. We denote this transformation by f tensor g.<br />
<br />
3b. Suppose V and W are finite dimensional. Show that Trace(f tensor g) = Trace(f) Trace(g). What does this say when f and g are both the identity transformation?<br />
<br />
4. Let V be a vector space. We define Sym^2 V to be the quotient of V tensor V by the subspace generated by all elements of the form<br />
<br />
v tensor w - w tensor v<br />
<br />
for v,w in V.<br />
<br />
Suppose dim V = n. What is dim Sym^2 V?<br />
<br />
5. Show that Z/pZ tensor Z/qZ is zero when p and q are distinct primes.<br />
<br />
==HOMEWORK 8 (due Nov 13)==<br />
<br />
1. (Schur's lemma can be false over non-algebraically closed fields) Let G be Z/3Z and let g be a generator of G. Then G acts by cyclic permutations on the set {1,2,3}; let P be the corresponding permutation representation on the REAL (not complex!) vector space generated by e_1,e_2,e_3.<br />
<br />
1a. Show that P breaks up as the direct sum of the trivial representation and a 2-dimensional representation V which is irreducible. (Remember, we are not working over the complex numbers, so you cannot use the criterion for irreducibility in terms of the character that we worked out in class for the complex case; you need to prove directly that V has no nontrivial proper subrepresentation.)<br />
<br />
1b. Show that the map from V to V induced by g is G-equivariant, but is neither zero nor a scalar multiplication.<br />
<br />
2. If V_1 and V_2 are representations (not necessarily irreducible!) of G, show that <chi_V_1, chi V_2> is a non-negative integer.<br />
<br />
3. When a formula always yields a non-negative integer, it is often because it is secretly a formula for either the cardinality of some finite set or the dimension of some finite-dimensional vector space. In this case, show that<br />
<br />
<chi_{V_1}, chi_{V_2}> = dim_C Hom_G(V_1,V_2)<br />
<br />
where we recall that Hom_G(V_1,G_2) refers to the space of G-equivariant homomorphisms from V_1 to V_2.<br />
<br />
4. If A is an abelian group, show that every irreducible representation of A is 1-dimensional. (Hint: show that there is a vector which is an eigenvector for every element of A!)<br />
<br />
5. Let G be the group of order 20 which is the semidirect product of A by H, where A is Z/5Z and H is (Z/5Z)^* with its natural action on A. We will work out all the irreducible representations of A. <br />
<br />
5a. First of all, consider the 5-dimensional permutation representation of G acting by left multiplication on the cosets of H. Show that this representation decomposes as a direct sum of the trivial representation and an irreducible 4-dimensional representation. (Hint: it is probably easier to use the character of this representation than to prove irreducibility directly.) <br />
<br />
5b. Now show that there are 4 1-dimensional representations of G which factor through the quotient G/A (i.e. such that rho(A) is the identity.)<br />
<br />
5c. Using the decomposition of the regular representation, show that the irreducible representations of G you have constructed are the only ones.<br />
<br />
==HOMEWORK 9 (due Nov 20)==<br />
<br />
0. Find all ring homomorphisms from Z to Z/30Z. [Warning: there are 8 of them, not 30.]<br />
<br />
<br />
1. If R is a ring (not necessarily with 1) such that a^2 = a for every a in R, R is called a Boolean ring.<br />
<br />
1a. Show that any nonzero Boolean ring has characteristic 2. (In other words, show that a+a=0 for all a in R.)<br />
<br />
1b. Show that every Boolean ring is commutative.<br />
<br />
<br />
2. Let phi : R -> S be a homomorphism of commutative rings with 1.<br />
<br />
2a. If I is an ideal of S, show that phi^(-1) (I) is an ideal of R.<br />
<br />
2b. If P is a prime ideal of S, show that phi^(-1) (P) is either R or a prime ideal of R.<br />
<br />
2c. If M is a maximal ideal of S and phi is surjective, show that phi^(-1) (M) is a maximal ideal of R.<br />
<br />
2d. Find a homomorphism phi : R -> S and a maximal ideal M of S such that phi^(-1) (M) is not a maximal ideal of R.<br />
<br />
<br />
3. Let Nil(A) denote the set of nilpotent elements of a ring A, and let R be a commutative ring. In class it was observed that Nil(R) is an ideal of R.<br />
<br />
3a. Find (a generator of) Nil(Z/720Z).<br />
<br />
3b. Show that Nil(R/Nil(R)) = 0.<br />
<br />
3c. If x is in Nil(R) and u is a unit, show that u+x is a unit.<br />
<br />
3d. Show that Nil(S), where S is the 2x2 matrices with real coefficients, is not an ideal of S.<br />
<br />
<br />
4a. If M is an R-module and I is an ideal of R such that im = 0 for all i in I and m in M, show that M carries the structure of an R/I-module.<br />
<br />
4b. If S is a commutative ring with 1 and J is a maximal ideal of S, show that J/J^2 is a vector space over S/J.<br />
<br />
<br />
5. A "divisible element" x of a Z-module is one such that for each positive integer k, there exists a y such that ky=x. Show that a free Z-module cannot contain a nonzero divisible element.<br />
<br />
<br />
6. Prove that the direct sum of any collection of free R-modules is also a free R-module.<br />
<br />
<br />
7. In this problem you will prove that the module M given by the direct product of copies of Z, indexed by the positive integers, is not a free Z-module. Therefore, assume that M is a free Z-module with basis B, and let N be the direct sum of copies of Z, as a submodule of M.<br />
<br />
7a. Show there is a countable subset B1 of B such that N is contained in the submodule N1 generated by B1. [Hint: N is countable.]<br />
<br />
7b. Show that M/N1 is a free Z-module.<br />
<br />
7c. Let S = {(b1, b2, b3, ...) : b_i = i! or -i!}. Show that there is some s in S which is not in N1. [Hint: S is uncountable.]<br />
<br />
7d. For any s in S but not N1, show that for each positive integer k, there exists some m in M with s-km in N1. Conclude that s+N1 is a divisible element of M/N1.<br />
<br />
7e. Use problem 5 to obtain a contradiction.<br />
<br />
<br />
8. In class I said "rank" is not always well-defined for R-modules if R is not commutative. Here is an example demonstrating this assertion. Let M be the countably infinite direct product of copies of Z as in exercise 7, and let R be the endomorphism ring of M. Define phi_1(a1,a2,a3,a4,...) = (a1,a3,a5,...) and phi_2(a1,a2,a3,a4,...) = (a2,a4,a6,...).<br />
<br />
8a. Show that {phi_1, phi_2} is a free basis for R, as a left R-module. [Hint: let psi_1(a1,a2,a3,...) = (a1,0,a2,0,...) and psi_2(a1,a2,a3,...) = (0,a1,0,a2,...). Compute the products of the phi_i and psi_j in either order, and use the result to show that the phi_i are a basis.]<br />
<br />
8b. Show that R is isomorphic to R^n for every positive integer n.<br />
<br />
<br />
==HOMEWORK 10 (due Dec 4)==<br />
<br />
1. '''How algebraists do calculus'''. The ''ring of dual numbers'' is the ring R = C[e]/(e^2). We think of e as an infinitesimal number; that is, something so small that its square is 0. This algebraic notion of infinitesimal is the basis for the algebraic notion of differentiation, which works as follows. Let f be a polynomial in R[x]. Show that there is a unique polynomial g in C[x] such that<br />
<br />
f(x+e) - f(x) = eg(x)<br />
<br />
and that in fact g is the derivative of f. In other words, the usual definition of "derivative" works just fine in this context, without any use of the notion of limit!<br />
<br />
2. Write down all the ideals of C[e]/e^2. (There are three.)<br />
<br />
3. Show that any right ideal of the ring M_2(Q) of 2x2 matrices is an ideal of the form I_V described in class for some subspace V of Q^2; that is, it consists precisely of those linear transformations whose image is contained in V.<br />
<br />
<br />
4. An ''idempotent'' element of a ring is an element e satisfying e^2 = e.<br />
<br />
4a. If e is an idempotent, show that 1-e is also an idempotent.<br />
<br />
4b. Suppose R has a unit (we are actually always supposing this, but I just want to remind you of it for this problem) and suppose e is a ''central'' idempotent, i.e. an idempotent which commutes with all elements of R. Show that eR and (1-e)R are both two-sided ideals. Moreover, show that eR and (1-e)R are also subrings of R (with the warning that the identity in eR is not 1, which is not contained in eR, but rather e.)<br />
<br />
4c. Show that the map f_e: R -> eR defined by f_e(r) = f(er) is a ring homomorphism. Finally show that the ring homomorphism<br />
<br />
f_{e} x f_{1-e}: R -> eR x (1-e)R <br />
<br />
is actually an isomorphism of rings. In other words, the presence of a central idempotent induces a decomposition a ring as a direct product.<br />
<br />
<br />
5. The ''center'' of a ring R is the subring of elements commuting with every element of R.<br />
<br />
5a. Show that the center of M_n(Q) is the ring of scalar matrices.<br />
<br />
5b. Show that the center of the group ring C[G] is the set of elements sum f(g) g such that f(g) is a class function.<br />
<br />
<br />
6. '''How algebraists do differential equations.''' We define the ''Weyl algebra'' W of differential operators in one variable as follows. Multiplication by x is a linear transformation on the complex vector space V of all smooth functions on C. Differentiation, which we denote d, is also a linear transformation on V. So we let W be the subring of End(V) generated by x and d. We note that x and d satisfy the relation xd = dx - 1; in particular, the Weyl algebra is not commutative.<br />
<br />
6a. Describe the submodule of V which is annihilated by the element d-x in W.<br />
<br />
6b. Show that, for any polynomial f, fd -df = f'. (This is another way algebraists do calculus!) Use this fact to show that the center of the Weyl algebra is just the field of constants. (Remark: you will use in the course of this proof the fact that a polynomial whose derivative is 0 is constant -- this would not be true if we were working in positive characteristic, as the example of x^p considered as a polynomial over F_p demonstrates.)<br />
<br />
<br />
7. If M is a left module for a ring R, we denote by Ann(M) the set of elements r in R such that rm = 0 for all m in M. Show that Ann(M) is a two-sided ideal. <br />
<br />
<br />
8. Let M be a left module for R, and let m be an element of M. We denote by Ann(m) the set of elements r in R such that rm = 0.<br />
<br />
8a. Show that Ann(m) is a left ideal, and show that the submodule Rv of M generated by v is isomorphic to the left module R/Ann(m).<br />
<br />
8c. Let V,W be as in question 6. Show that Ann(e^x) is the left ideal W(d-1).</div>Dummithttps://hilbert.math.wisc.edu/wiki/index.php?title=741&diff=47417412012-11-26T03:36:32Z<p>Dummit: /* HOMEWORK 9 (due Nov 20) */ added extra part for clarity</p>
<hr />
<div>'''Math 741'''<br />
<br />
Algebra<br />
<br />
Prof: [http://www.math.wisc.edu/~ellenber Jordan Ellenberg]<br />
<br />
Grader: Evan Dummit<br />
<br />
Ellenberg's office hours: Friday 3pm.<br />
<br />
Grader's office hours: Monday 4pm. Late homework may be given directly to the grader, along with either (i) the instructor's permission, or (ii) a polite request for mercy.<br />
<br />
This course, the first semester of the introductory graduate sequence in algebra, will cover the basic theory of groups, group actions, representation, linear and multilinear algebra, and the beginnings of ring theory.<br />
<br />
==SYLLABUS==<br />
<br />
In this space we will record the theorems and definitions we covered each week, which we can use as a list of notions you should be prepared to answer questions about on the Algebra qualifying exam. The material covered on the homework is also an excellent guide to the scope of the course. <br />
<br />
'''WEEK 1''': <br />
<br />
Definition of group. Associativity. Inverse. <br />
<br />
Examples of group: GL_n(R). GL_n(Z). Z/nZ. R. Z. R^*. The free group F_k on k generators. <br />
<br />
Homomorphisms. The homomorphisms from F_k to G are in bijection with G^k. Isomorphisms.<br />
<br />
'''WEEK 2''':<br />
<br />
The symmetric group (or permutation group) S_n on n letters. Cycle decomposition of a permutation. Order of a permutation. Thm: every element of a finite group has finite order. <br />
<br />
Subgroups. Left and right cosets. Lagrange's Theorem. Cyclic groups. The order of an element of a finite group is a divisor of the order of the group.<br />
<br />
The sign homomorphism S_n -> +-1.<br />
<br />
<br />
'''WEEK 3'''<br />
<br />
Normal subgroups. The quotient of a group by a normal subgroup. The first isomorphism theorem. Examples of S_n -> +-1 and S_4 -> S_3 with kernel V_4, the Klein 4-group.<br />
<br />
Centralizers and centers. Abelian groups. The center of SL_n(R) is either 1 or +-1.<br />
<br />
Groups with presentations. The infinite dihedral group <x,y | x^2 = 1, y^2 = 1>.<br />
<br />
'''WEEK 4'''<br />
<br />
More on groups with presentations.<br />
<br />
Second and third isomorphism theorems.<br />
<br />
Semidirect products.<br />
<br />
'''WEEK 5'''<br />
<br />
Group actions, orbits, and stabilizers.<br />
<br />
Orbit-stabilizer theorem.<br />
<br />
Cayley's theorem.<br />
<br />
Cauchy's theorem.<br />
<br />
'''WEEK 6'''<br />
<br />
Applications of orbit-stabilizer theorem (p-groups have nontrivial center, first Sylow theorem.)<br />
<br />
Classification of finite abelian groups and finitely generated abelian groups.<br />
<br />
Composition series and the Jordan-Holder theorem (which we state but don't prove.) <br />
<br />
The difference between knowing the composition factors and knowing the group (e.g. all p-groups of the same order have the same composition factors.)<br />
<br />
'''WEEK 7'''<br />
<br />
Simplicity of A_n.<br />
<br />
Nilpotent groups (main example: the Heisenberg group)<br />
<br />
Derived series and lower central series.<br />
<br />
Category theory 101: Definition of category and functor. Some examples. A group is a groupoid with one object.<br />
<br />
'''WEEK 8'''<br />
<br />
Introduction to representation theory.<br />
<br />
'''WEEK 10'''<br />
<br />
Ring theory 101: Rings, ring homomorphisms, ideals, isomorphism theorems. Examples: fields, Z, the Hamilton quaternions, matrix rings, rings of polynomials and formal power series, quadratic integer rings, group rings. Integral domains. Maximal and prime ideals. The nilradical.<br />
<br />
Module theory 101: Modules, module homomorphisms, submodules, isomorphism theorems. Noetherian modules and Zorn's lemma. Direct sums and direct products of arbitrary collections of modules.<br />
<br />
<br />
Below you will find a repository of homework problems. Note that some of these problems are taken from Lang's ''Algebra''.<br />
<br />
==HOMEWORK 1 (due Sep 20)==<br />
<br />
1. Suppose that H_1 and H_2 are subgroups of a group G. Prove that the intersection of H_1 and H_2 is a subgroup of G.<br />
<br />
2. Recall that S_3 (the symmetric group) is the group of permutations of the set {1..3}. List all the subgroups of S_3.<br />
<br />
3. We can define an equivalence relation on rational numbers by declaring two rational numbers to be equal whenever they differ by an integer. We denote the set of equivalence classes by Q/Z. The operation of addition makes Q/Z into a group.<br />
<br />
a) For each n, prove that Q/Z has a subgroup of order n.<br />
<br />
b) Prove that Q/Z is a ''divisible'' group: that is, if x is an element of Q/Z and n is an integer, there exists an element y of Q/Z such that ny = x. (Note that we write the operation in this group as addition rather than multiplication, which is why we write ny for the n-fold product of y with itself rather than y^n)<br />
<br />
c) Prove that Q/Z is not finitely generated. (Hint: prove that if x_1, .. x_d is a finite subset of Q/Z, the subgroup of Q/Z generated by x_1, ... x_d is finite.)<br />
<br />
4. Let F_2 be the free group on two generators x,y. Prove that there is a automorphism of F_2 which sends x to xyx and y to xy, and prove that this automorphism is unique.<br />
<br />
5. Let H be a subgroup of G, and let N_G(H), the ''normalizer'' of H in G, be the set of elements g in G satsifying g H g^{-1} = H. Prove that N_G(H) is a subgroup of G.<br />
<br />
6. Let T be the subgroup of GL_2(R) consisting of diagonal matrices. This is an example of a "Cartan subgroup," or "torus" (whence the notation T.) Describe the normalizer N(T) of T in GL_2(R). Prove that there is a homomorphism from N(T) to Z/2Z whose kernel is precisely T. <br />
<br />
7. Prove that the group of inner automorphisms of a group is normal in the full automorphism group Aut(G).<br />
<br />
8. Let H and H' be subgroups of a group G, and let x be an element of G. We denote by HxH' the set of elements of the form hxh', with h in H and h' in H', and we call HxH a ''double coset'' of the pair (H,H').<br />
<br />
a) Show that G decomposes as a disjoint union of double cosets of (H,H').<br />
<br />
b) Let G be GL_2(R) and let B be the subgroup of upper triangular matrices. Prove that G decomposes into exactly two double cosets of (B,B).<br />
<br />
c) Let G be the symmetric group S_n, and let H be the subgroup of permutations which fix 1. Describe the double coset decomposition of S_n into double cosets of (H,H).<br />
<br />
==HOMEWORK 2 (due Sep 27)==<br />
<br />
1. Prove that the symmetric group S_n is generated by transpositions. Hint: proceed by induction on n. Let H be the subgroup of S_n consisting of permutations fixing 1. Observe that H is isomorphic to S_{n-1}, so that H is generated by transpositions. Explain why it suffices to show that, for any g in S_n, the coset gH contains a transposition. Then explain why this is the case.<br />
<br />
2. Given that the symmetric group S_n is generated by transpositions, show that there are no nontrivial homomorphisms from S_n to Z/pZ for any odd prime p.<br />
<br />
3. Compute the centralizer of the triple flip (12)(34)(56) in S_6. (Note: this is the kind of problem that a computer algebra package like sage is very good at, and I encourage you to learn to use sage well enough to check your answer.)<br />
<br />
4. Warning: a normal subgroup of a normal subgroup need not be normal! Let H be the subgroup of S_4 generated by (12)(34). Then H is a subgroup of the Klein 4-group V_4, which we have already shown is normal in S_4. Show that H is normal in V_4, but H is not normal in S_4.<br />
<br />
5. The '''quaternion group''' Q is a non-abelian finite group of order 8. Its elements are 1, -1, i, -i, j, -j, k, -k; it satisfies g^2 = -1 for all g except +-1, and ijk = 1. (Note that the notation "-i" means "the product of i with -1" and -1 is understood to be central in Q.) Prove that every subgroup of Q is normal.<br />
<br />
6. (finishing example done in class) Let Gamma be the group F<x,y> / (x^2 = 1, y^2 = 1). (Remember, this means the quotient of the free group by the group generated by all conjugates of x^2 and y^2.) Prove that Gamma is infinite. Prove that Gamma has a subgroup R generated by xy isomorphic to Z, that R is normal, and that the quotient Gamma/R is isomorphic to Z/2Z. <br />
<br />
7. Suppose a group G has the property that every element in g satisfies g^2 = 1. Show that G must be abelian.<br />
<br />
==HOMEWORK 3 (due Oct 4)==<br />
<br />
1. Multiplication by -1 is an automorphism of Z/nZ. Let a be the homomorphism Z/2Z -> Aut(Z/nZ) which sends the nontrivial element of Z/2Z to multiplication by -1. Then we have a semidirect product of Z/pZ by Z/2Z as defined in class. This group of order 2n is called a ''dihedral group.'' and is denoted D_n (or sometimes D_{2n}).<br />
<br />
1a. Compute the center of D_n. (Note that the answer depends on n!)<br />
1b. Show that all involutions of D_n are conjugate to each other if and only if n is odd.<br />
<br />
2. (More on direct products that we didn't do in class.) Suppose that G is a semidirect product of N with H. Suppose furthermore that H is also normal in G. Prove that G is a direct product of N with H.<br />
<br />
3. Let Q be the quaternion group from last week's homework. Show that Q does not decompose as a semidirect product N \rtimes H, apart form the trivial cases N = {e} and N = Q.<br />
<br />
4. The ''affine linear group'' of degree n is the group of transformations from R^n to R^n of the form x -> Ax + b, where A is a matrix in GL_n(R) and b is a vector in R^n. Prove that the affine linear group is a semidirect product R^n \rtimes GL_n.<br />
<br />
5. The ''ordinary triangle group'' T(p,q,r) is the group with presentation <x,y | x^p = y^q = (xy)^r = 1>. This is a very interesting family of groups which arises naturally in many contexts in number theory and topology.<br />
<br />
6a. Prove that T(2,2,n) is isomorphic to the dihedral group D_n.<br />
6b. Prove that T(2,3,3) is finite and compute its order. Is it isomorphic to a finite group we've discussed in class?<br />
<br />
(It is a beautiful fact, but outside the scope of this course, that T(p,q,r) is finite if and only if 1/p + 1/q + 1/r > 1.)<br />
<br />
7. Let G be a subgroup of GL_n(R) which contains SL_n(R). Prove that G is a normal subgroup of GL_n(R).<br />
<br />
8. Let G be a group and let H_1 and H_2 be subgroups of G of finite index (that is, the number of cosets in G/H_i is finite, i = 1,2.) Prove that H_1 intersect H_2 also has finite index in G.<br />
<br />
9. Recall that the commutator [x,y] is x y x^{-1} y^{-1}. If G is a group, the commutator subgroup of G, denoted G' or [G,G], is the subgroup of G generated by all commutators [x,y] for x,y in G.<br />
<br />
9a. Show that G' is a normal subgroup of G.<br />
9b. Show that G/G' is an abelian group.<br />
9c. Show that if f: G -> A is a homomorphism, with A abelian, then G' is contained in ker f. Conclude that f factors through a homomorphism G/G' -> A.<br />
9d. Show that G' = G if and only if G has no nontrivial abelian quotient. In this case, we say G is ''perfect''.<br />
<br />
10. Under what conditions on (p,q,r) is the triangle group T(p,q,r) perfect?<br />
<br />
==HOMEWORK 4 (due Oct 16)==<br />
<br />
1. Let X be a set with a transitive action of a group G, which we think of as a homomorphism f: G -> Sym(X). Let H_x be the stabilizer of an element x of X.<br />
<br />
1a. If x' is another element of X, show that H_{x"} and H_x are conjugate subgroups. (Hint: this is not true without the hypothesis of transitivity, so make sure to use it!)<br />
<br />
1b. Show that the kernel of f is the intersection of all the conjugates of H_x in G. (This subgroup is called the ''normal core'' of H_x.)<br />
<br />
2. This week, a certified mathematical grown-up asked on MathOverflow what the index-3 subgroups of SL_2(Z/pZ) were. Let's do it ourselves for homework. Let p be a prime, and suppose H is an index-3 subgroup of SL_2(Z/pZ). I do not claim that the proof sketched below is necessarily the easiest, but it allows us to cover some ground that we won't get to do in lecture.<br />
<br />
2a. Using the coset action, show that the normal core of H is the kernel of a homomorphism f from SL_2(Z/pZ) to S_3.<br />
<br />
2b. A ''unipotent matrix'' is a matrix A such that (A-I)^n = 0 for some positive integer n. Prove that a unipotent element of SL_2(Z/pZ) has order dividing p.<br />
<br />
2c. You may use the fact that unipotent matrices generate SL_2(Z/pZ). Given this fact, show that f must be trivial when p > 3.<br />
<br />
2d. Using the above, show that SL_2(Z/pZ) has no index-3 subgroups when p > 3.<br />
<br />
2e. (extra credit) What can you say about the minimal index of a proper subgroup of SL_2(Z/pZ)?<br />
<br />
3. Let H be a subgroup of G of index 2. Prove that H is normal.<br />
<br />
4. We proved two theorems in class: Lagrange's theorem, which says that the order of a subgroup of G divides |G|, and Cauchy's theorem, which says that if p divides |G| then there exists a subgroup of G of order p. Cauchy's theorem is a kind of partial converse to Lagrange's theorem, but the full converse is false. Give an example of a finite group G and a divisor n of |G| such that you can prove there is no subgroup of G of order n.<br />
<br />
5. Let X be the set of lines in F_3^2 (here a line means a 1-dimensional linear subspace) so that |X| = 4. Let G be the projective linear group PGL_2(F_3), so that G acts on X via its action on F_3^2. This action can be thought of as a homomorphism from G to S_4. Show that this homomorphism is an ISOMORPHISM from PGL_2(F_3) to S_4.<br />
<br />
6. The group S_3 has only three conjugacy classes. Prove that a finite group with at most three conjugacy classes has order at most 6.<br />
<br />
7. Let X be the set of ordered triples of elements of {1,..,n}, for some n >= 3. Then S_n acts on X. How many orbits are there? (Hint: the answer does not depend on n.)<br />
<br />
8. Suppose that G and H are groups and f: G -> H is a homomorphism. Recall the definition of the abelianization G^ab from the previous problem set.<br />
<br />
8a. Show that the composition G -> H -> H^ab factors through a unique homomorphism G^ab -> H^ab, which we denote f^ab.<br />
<br />
8b. Show that if f: G -> H and g: H -> Q are homomorphisms, then f^ab o g^ab = (f o g)^ab.<br />
<br />
(For those reading MacLane, this constitutes a proof that abelianization is a '''functor''' from the category of groups to the category of abelian groups.)<br />
<br />
==HOMEWORK 5 (due Oct 23)==<br />
<br />
1. Show that there are only two different isomorphism classes of groups of order 14. (Hint: prove that such a group must be a semidirect product of a group of order 7 by a group of order 2.)<br />
<br />
2. (variant of Burnside's Lemma) Let G be a finite group and H a normal subgroup. Suppose G acts on a finite set X. <br />
<br />
2a. Show that if x and y lie in the same H-orbit, and g is an element of G, then gx and gy also lie in the same H-orbit. Thus, G/H has a natural action on the set of H-orbits X/H.<br />
<br />
2b. Show that the average number of fixed points in X of an element of the coset gH is the same as the number of fixed points of the element gH of G/H in its action on X/H.<br />
<br />
3. A ''central extension'' of a group G by an abelian group A is a group E, together with a surjective homomorphism E -> G whose kernel is central in E and is isomorphic to A.<br />
<br />
3a. Show that the quaternion group is a central extension of (Z/2Z)^2 by Z/2Z.<br />
<br />
3b. Give an example of another nonabelian group which is a central extension of (Z/2Z)^2 by Z/2Z and which is not isomorphic to the quaternion group.<br />
<br />
4. Prove that a central extension of an abelian group is nilpotent.<br />
<br />
5. Give two different composition series for S_4 and show that they have the same composition factors.<br />
<br />
6. Suppose that A is a finitely generated abelian group, and B is a subgroup of A.<br />
Prove that rank(A/B) = rank(A) - rank(B).<br />
ADDED: I asserted in class, but did not prove, that a subgroup of a finitely generated abelian group is again finitely generated. As one class member pointed out, without this fact it's not even clear why it makes sense to talk about rank(B)! You may assume it (though it would have been better for me to have included proving this fact as part of the problem....)<br />
<br />
7. For this problem, you may use the fact (which I should have asked on homework earlier) that if H_1 and H_2 are subgroups of G, then<br />
<br />
[G: H_1 intersect H_2] <= [G:H_1][G:H_2].<br />
<br />
Prove that if G is a simple group, d is an integer greater than 1, and S is a subgroup of index d in G, then |G| <= d^d.<br />
<br />
Conclude that, in particular, A_5 has no subgroup of index 3. On the other hand, exhibit a subgroup of A_5 of index 5.<br />
<br />
(Note: d^d is not a sharp bound; can you get a bound of d factorial instead?)<br />
<br />
<!--(This argument shows that a solvable subgroup of A_n has size at most about (n-1)!, but this is far from sharp; e.g. I found a citation to a 1967 paper of Dixon which shows that a solvable subgroup of A_n has size at most k^n where k is around 2.88.)--><br />
<br />
8. Give an example of an abelian group A with a subgroup B such that B is not a direct summand of A.<br />
<br />
9. Let A be the subgroup of Z^2 generated by the vector (1,0). Let B be the subgroup of Z^2 generated by the vector (3,3). Show that A and B are isomorphic to each other, but Z^2 / A is not isomorphic to Z^2 / B.<br />
<br />
==HOMEWORK 6 (due Oct 30)==<br />
<br />
1. In this problem, you will construct a functor F from the category of finite sets to the category of complex vector spaces. I'll tell you what it does on objects: if X is a finite set, then F(X) is the vector space spanned by a set of basis elements {e_x}_{x in X}, so that dim F(X) = |X|. Your job: complete the definition of F by describing the map from Mor_{Finite Sets}(X,Y) to Mor_{Vector Spaces}(F(X),F(Y)), and showing that your map respects composition.<br />
<br />
2. If (V, rho) is a representation of a group G, denote by V^G the space of vectors in V such that gv = v for all g in G; these are called _invariant_ vectors. We saw that the permutation representation of S_3 has a one-dimensional space of invariants. Prove that if X is a finite set with G-action, and V_X the corresponding permutation representation of G, then<br />
<br />
dim V_X^G = number of orbits of X.<br />
<br />
3. (Invariants are functorial) Let G be a group. Then there is a category called C[G]-Mod whose objects are complex representations of G (i.e. pairs (V,rho) where V is a complex vector space and rho is a homomorphism from G to GL(V)). The morphisms in this category are the ones described in class: Mor((V,rho),(W,psi)) is the set of linear maps f from V to W such that<br />
<br />
f(rho(g)(v)) = psi(g)(f(v))<br />
<br />
for all g in G and all v in V.<br />
<br />
Show that there is a functor H_0 from the category of representations of G to the category of vector spaces whose action on objects is given by<br />
<br />
H_0((V) = V^G.<br />
<br />
(In other words: your job again is to explain how to associate to a morphism from (V,rho) to (W,phi) a morphism from V^G to W^G, in a way that's compatible with composition.)<br />
<br />
<!--4. The group PSL_2(Z) has a presentation of the form [U,V|U^2 = V^3 = 1]. Thus, an n-dimensional representation of PSL_2(Z) is precisely a pair of n x n matrices (rho(U),rho(V)), such that rho(U)^2 = rho(V)^3 = 1. Prove that for every n > 1 there are uncountably many isomorphism classes of n-dimensional representations of PSL_2(Z). (Extra credit: show that there are uncountably many isomorphism classes of _irreducible_ representations.)<br />
<br />
An amazing theorem of Margulis (superrigidity) shows that SL_2(Z) is very special among arithmetic groups in this way. For instance, there are only _finitely_ many isomorphism classes of n-dimensional representations of PSL_d(Z) for any d > 2; those groups are "rigid."--><br />
<br />
5. Let G be a finite group, let X be a set with G-action, and let chi be the character of the corresponding permutation representation. Show that, for each g in G, chi(g) is the number of elements of X fixed by g. (I am actually going to prove this in class.)<br />
<br />
6. Let V be the space of homogeneous degree-3 polynomials in variables x_1, x_2, x_3; then S_3 acts on V by permutation of the three variables. Describe the decomposition of V into irreducible representations of S_3.<br />
<br />
7a. Show that if chi is the character of a representation of a finite group G, then chi(g^{-1}) is the complex conjugate of chi.<br />
<br />
7b. As a consequence, show that if g is conjugate to g^{-1} for all g in G, then chi(g) is real.<br />
<br />
7c. The groups D_p and S_n have the property that g is conjugate to g^{-1} for _every_ g in G, so that every character is a real-valued function on G. Give an example of a group G where this is not the case, and give an example of a representation of G whose character takes non-real values.<br />
<br />
==HOMEWORK 7 (due Nov 6)==<br />
<br />
1. Let G be a finite group, let X be a finite set with G-action, and let chi be the character of the corresponding permutation representation. Show that, for each g in G, chi(g) is the number of elements of X fixed by g. (I am actually going to prove this in class.)<br />
<br />
2a. Show that if chi is the character of a finite-dimensional representation of a finite group G, then chi(g^{-1}) is the complex conjugate of chi(g) for all g.<br />
<br />
2b. As a consequence, show that if g is conjugate to g^{-1} for all g in G, then chi(g) is real.<br />
<br />
2c. The groups D_p and S_n have the property that g is conjugate to g^{-1} for _every_ g in G, so that every character is a real-valued function on G. Give an example of a group G where this is not the case, and give an example of a representation of G whose character takes non-real values.<br />
<br />
3. Let V and W be complex vector spaces. Let f be a linear transformation of V, and g a linear transformation of W. <br />
<br />
3a. Show that there is a unique linear transformation F satisfying<br />
<br />
F(v tensor w) = f(v) tensor g(w)<br />
<br />
for all v in V and all w in W. We denote this transformation by f tensor g.<br />
<br />
3b. Suppose V and W are finite dimensional. Show that Trace(f tensor g) = Trace(f) Trace(g). What does this say when f and g are both the identity transformation?<br />
<br />
4. Let V be a vector space. We define Sym^2 V to be the quotient of V tensor V by the subspace generated by all elements of the form<br />
<br />
v tensor w - w tensor v<br />
<br />
for v,w in V.<br />
<br />
Suppose dim V = n. What is dim Sym^2 V?<br />
<br />
5. Show that Z/pZ tensor Z/qZ is zero when p and q are distinct primes.<br />
<br />
==HOMEWORK 8 (due Nov 13)==<br />
<br />
1. (Schur's lemma can be false over non-algebraically closed fields) Let G be Z/3Z and let g be a generator of G. Then G acts by cyclic permutations on the set {1,2,3}; let P be the corresponding permutation representation on the REAL (not complex!) vector space generated by e_1,e_2,e_3.<br />
<br />
1a. Show that P breaks up as the direct sum of the trivial representation and a 2-dimensional representation V which is irreducible. (Remember, we are not working over the complex numbers, so you cannot use the criterion for irreducibility in terms of the character that we worked out in class for the complex case; you need to prove directly that V has no nontrivial proper subrepresentation.)<br />
<br />
1b. Show that the map from V to V induced by g is G-equivariant, but is neither zero nor a scalar multiplication.<br />
<br />
2. If V_1 and V_2 are representations (not necessarily irreducible!) of G, show that <chi_V_1, chi V_2> is a non-negative integer.<br />
<br />
3. When a formula always yields a non-negative integer, it is often because it is secretly a formula for either the cardinality of some finite set or the dimension of some finite-dimensional vector space. In this case, show that<br />
<br />
<chi_{V_1}, chi_{V_2}> = dim_C Hom_G(V_1,V_2)<br />
<br />
where we recall that Hom_G(V_1,G_2) refers to the space of G-equivariant homomorphisms from V_1 to V_2.<br />
<br />
4. If A is an abelian group, show that every irreducible representation of A is 1-dimensional. (Hint: show that there is a vector which is an eigenvector for every element of A!)<br />
<br />
5. Let G be the group of order 20 which is the semidirect product of A by H, where A is Z/5Z and H is (Z/5Z)^* with its natural action on A. We will work out all the irreducible representations of A. <br />
<br />
5a. First of all, consider the 5-dimensional permutation representation of G acting by left multiplication on the cosets of H. Show that this representation decomposes as a direct sum of the trivial representation and an irreducible 4-dimensional representation. (Hint: it is probably easier to use the character of this representation than to prove irreducibility directly.) <br />
<br />
5b. Now show that there are 4 1-dimensional representations of G which factor through the quotient G/A (i.e. such that rho(A) is the identity.)<br />
<br />
5c. Using the decomposition of the regular representation, show that the irreducible representations of G you have constructed are the only ones.<br />
<br />
==HOMEWORK 9 (due Nov 20)==<br />
<br />
0. Find all ring homomorphisms from Z to Z/30Z. [Warning: there are 8 of them, not 30.]<br />
<br />
<br />
1. If R is a ring (not necessarily with 1) such that a^2 = a for every a in R, R is called a Boolean ring.<br />
<br />
1a. Show that any nonzero Boolean ring has characteristic 2. (In other words, show that a+a=0 for all a in R.)<br />
<br />
1b. Show that every Boolean ring is commutative.<br />
<br />
<br />
2. Let phi : R -> S be a homomorphism of commutative rings with 1.<br />
<br />
2a. If I is an ideal of S, show that phi^(-1) (I) is an ideal of R.<br />
<br />
2b. If P is a prime ideal of S, show that phi^(-1) (P) is either R or a prime ideal of R.<br />
<br />
2c. If M is a maximal ideal of S and phi is surjective, show that phi^(-1) (M) is a maximal ideal of R.<br />
<br />
2d. Find a homomorphism phi : R -> S and a maximal ideal M of S such that phi^(-1) (M) is not a maximal ideal of R.<br />
<br />
<br />
3. Let Nil(A) denote the set of nilpotent elements of a ring A, and let R be a commutative ring. In class it was observed that Nil(R) is an ideal of R.<br />
<br />
3a. Find (a generator of) Nil(Z/720Z).<br />
<br />
3b. Show that Nil(R/Nil(R)) = 0.<br />
<br />
3c. If x is in Nil(R) and u is a unit, show that u+x is a unit.<br />
<br />
3d. Show that Nil(S), where S is the 2x2 matrices with real coefficients, is not an ideal of S.<br />
<br />
<br />
4a. If M is an R-module and I is an ideal of R such that im = 0 for all i in I and m in M, show that M carries the structure of an R/I-module.<br />
<br />
4b. If S is a commutative ring with 1 and J is a maximal ideal of S, show that J/J^2 is a vector space over S/J.<br />
<br />
<br />
5. A "divisible element" x of a Z-module is one such that for each positive integer k, there exists a y such that ky=x. Show that a free Z-module cannot contain a nonzero divisible element.<br />
<br />
<br />
6. Prove that the direct sum of any collection of free R-modules is also a free R-module.<br />
<br />
<br />
7. In this problem you will prove that the module M given by the direct product of copies of Z, indexed by the positive integers, is not a free Z-module. Therefore, assume that M is a free Z-module with basis B, and let N be the direct sum of copies of Z, as a submodule of M.<br />
<br />
7a. Show there is a countable subset B1 of B such that N is contained in the submodule N1 generated by B1. [Hint: N is countable.]<br />
<br />
7b. Show that M/N1 is a free Z-module.<br />
<br />
7c. Let S = {(b1, b2, b3, ...) : b_i = i! or -i!}. Show that there is some s in S which is not in N1. [Hint: S is uncountable.]<br />
<br />
7d. For any s in S but not N1, show that for each positive integer k, there exists some m in M with s-km in N1. Conclude that s+N1 is a divisible element of M/N1.<br />
<br />
7e. Use problem 5 to obtain a contradiction.<br />
<br />
<br />
8. In class I said "rank" is not always well-defined for R-modules if R is not commutative. Here is an example demonstrating this assertion. Let M be the countably infinite direct product of copies of Z as in exercise 7, and let R be the endomorphism ring of M. Define phi_1(a1,a2,a3,a4,...) = (a1,a3,a5,...) and phi_2(a1,a2,a3,a4,...) = (a2,a4,a6,...).<br />
<br />
8a. Show that {phi_1, phi_2} is a free basis for R, as a left R-module. [Hint: let psi_1(a1,a2,a3,...) = (a1,0,a2,0,...) and psi_2(a1,a2,a3,...) = (0,a1,0,a2,...). Compute the products of the phi_i and psi_j in either order, and use the result to show that the phi_i are a basis.]<br />
<br />
8b. Show that R is isomorphic to R^n for every positive integer n.</div>Dummithttps://hilbert.math.wisc.edu/wiki/index.php?title=741&diff=47407412012-11-26T01:27:29Z<p>Dummit: /* HOMEWORK 9 (due Nov 20) */ added problem for future sets</p>
<hr />
<div>'''Math 741'''<br />
<br />
Algebra<br />
<br />
Prof: [http://www.math.wisc.edu/~ellenber Jordan Ellenberg]<br />
<br />
Grader: Evan Dummit<br />
<br />
Ellenberg's office hours: Friday 3pm.<br />
<br />
Grader's office hours: Monday 4pm. Late homework may be given directly to the grader, along with either (i) the instructor's permission, or (ii) a polite request for mercy.<br />
<br />
This course, the first semester of the introductory graduate sequence in algebra, will cover the basic theory of groups, group actions, representation, linear and multilinear algebra, and the beginnings of ring theory.<br />
<br />
==SYLLABUS==<br />
<br />
In this space we will record the theorems and definitions we covered each week, which we can use as a list of notions you should be prepared to answer questions about on the Algebra qualifying exam. The material covered on the homework is also an excellent guide to the scope of the course. <br />
<br />
'''WEEK 1''': <br />
<br />
Definition of group. Associativity. Inverse. <br />
<br />
Examples of group: GL_n(R). GL_n(Z). Z/nZ. R. Z. R^*. The free group F_k on k generators. <br />
<br />
Homomorphisms. The homomorphisms from F_k to G are in bijection with G^k. Isomorphisms.<br />
<br />
'''WEEK 2''':<br />
<br />
The symmetric group (or permutation group) S_n on n letters. Cycle decomposition of a permutation. Order of a permutation. Thm: every element of a finite group has finite order. <br />
<br />
Subgroups. Left and right cosets. Lagrange's Theorem. Cyclic groups. The order of an element of a finite group is a divisor of the order of the group.<br />
<br />
The sign homomorphism S_n -> +-1.<br />
<br />
<br />
'''WEEK 3'''<br />
<br />
Normal subgroups. The quotient of a group by a normal subgroup. The first isomorphism theorem. Examples of S_n -> +-1 and S_4 -> S_3 with kernel V_4, the Klein 4-group.<br />
<br />
Centralizers and centers. Abelian groups. The center of SL_n(R) is either 1 or +-1.<br />
<br />
Groups with presentations. The infinite dihedral group <x,y | x^2 = 1, y^2 = 1>.<br />
<br />
'''WEEK 4'''<br />
<br />
More on groups with presentations.<br />
<br />
Second and third isomorphism theorems.<br />
<br />
Semidirect products.<br />
<br />
'''WEEK 5'''<br />
<br />
Group actions, orbits, and stabilizers.<br />
<br />
Orbit-stabilizer theorem.<br />
<br />
Cayley's theorem.<br />
<br />
Cauchy's theorem.<br />
<br />
'''WEEK 6'''<br />
<br />
Applications of orbit-stabilizer theorem (p-groups have nontrivial center, first Sylow theorem.)<br />
<br />
Classification of finite abelian groups and finitely generated abelian groups.<br />
<br />
Composition series and the Jordan-Holder theorem (which we state but don't prove.) <br />
<br />
The difference between knowing the composition factors and knowing the group (e.g. all p-groups of the same order have the same composition factors.)<br />
<br />
'''WEEK 7'''<br />
<br />
Simplicity of A_n.<br />
<br />
Nilpotent groups (main example: the Heisenberg group)<br />
<br />
Derived series and lower central series.<br />
<br />
Category theory 101: Definition of category and functor. Some examples. A group is a groupoid with one object.<br />
<br />
'''WEEK 8'''<br />
<br />
Introduction to representation theory.<br />
<br />
'''WEEK 10'''<br />
<br />
Ring theory 101: Rings, ring homomorphisms, ideals, isomorphism theorems. Examples: fields, Z, the Hamilton quaternions, matrix rings, rings of polynomials and formal power series, quadratic integer rings, group rings. Integral domains. Maximal and prime ideals. The nilradical.<br />
<br />
Module theory 101: Modules, module homomorphisms, submodules, isomorphism theorems. Noetherian modules and Zorn's lemma. Direct sums and direct products of arbitrary collections of modules.<br />
<br />
<br />
Below you will find a repository of homework problems. Note that some of these problems are taken from Lang's ''Algebra''.<br />
<br />
==HOMEWORK 1 (due Sep 20)==<br />
<br />
1. Suppose that H_1 and H_2 are subgroups of a group G. Prove that the intersection of H_1 and H_2 is a subgroup of G.<br />
<br />
2. Recall that S_3 (the symmetric group) is the group of permutations of the set {1..3}. List all the subgroups of S_3.<br />
<br />
3. We can define an equivalence relation on rational numbers by declaring two rational numbers to be equal whenever they differ by an integer. We denote the set of equivalence classes by Q/Z. The operation of addition makes Q/Z into a group.<br />
<br />
a) For each n, prove that Q/Z has a subgroup of order n.<br />
<br />
b) Prove that Q/Z is a ''divisible'' group: that is, if x is an element of Q/Z and n is an integer, there exists an element y of Q/Z such that ny = x. (Note that we write the operation in this group as addition rather than multiplication, which is why we write ny for the n-fold product of y with itself rather than y^n)<br />
<br />
c) Prove that Q/Z is not finitely generated. (Hint: prove that if x_1, .. x_d is a finite subset of Q/Z, the subgroup of Q/Z generated by x_1, ... x_d is finite.)<br />
<br />
4. Let F_2 be the free group on two generators x,y. Prove that there is a automorphism of F_2 which sends x to xyx and y to xy, and prove that this automorphism is unique.<br />
<br />
5. Let H be a subgroup of G, and let N_G(H), the ''normalizer'' of H in G, be the set of elements g in G satsifying g H g^{-1} = H. Prove that N_G(H) is a subgroup of G.<br />
<br />
6. Let T be the subgroup of GL_2(R) consisting of diagonal matrices. This is an example of a "Cartan subgroup," or "torus" (whence the notation T.) Describe the normalizer N(T) of T in GL_2(R). Prove that there is a homomorphism from N(T) to Z/2Z whose kernel is precisely T. <br />
<br />
7. Prove that the group of inner automorphisms of a group is normal in the full automorphism group Aut(G).<br />
<br />
8. Let H and H' be subgroups of a group G, and let x be an element of G. We denote by HxH' the set of elements of the form hxh', with h in H and h' in H', and we call HxH a ''double coset'' of the pair (H,H').<br />
<br />
a) Show that G decomposes as a disjoint union of double cosets of (H,H').<br />
<br />
b) Let G be GL_2(R) and let B be the subgroup of upper triangular matrices. Prove that G decomposes into exactly two double cosets of (B,B).<br />
<br />
c) Let G be the symmetric group S_n, and let H be the subgroup of permutations which fix 1. Describe the double coset decomposition of S_n into double cosets of (H,H).<br />
<br />
==HOMEWORK 2 (due Sep 27)==<br />
<br />
1. Prove that the symmetric group S_n is generated by transpositions. Hint: proceed by induction on n. Let H be the subgroup of S_n consisting of permutations fixing 1. Observe that H is isomorphic to S_{n-1}, so that H is generated by transpositions. Explain why it suffices to show that, for any g in S_n, the coset gH contains a transposition. Then explain why this is the case.<br />
<br />
2. Given that the symmetric group S_n is generated by transpositions, show that there are no nontrivial homomorphisms from S_n to Z/pZ for any odd prime p.<br />
<br />
3. Compute the centralizer of the triple flip (12)(34)(56) in S_6. (Note: this is the kind of problem that a computer algebra package like sage is very good at, and I encourage you to learn to use sage well enough to check your answer.)<br />
<br />
4. Warning: a normal subgroup of a normal subgroup need not be normal! Let H be the subgroup of S_4 generated by (12)(34). Then H is a subgroup of the Klein 4-group V_4, which we have already shown is normal in S_4. Show that H is normal in V_4, but H is not normal in S_4.<br />
<br />
5. The '''quaternion group''' Q is a non-abelian finite group of order 8. Its elements are 1, -1, i, -i, j, -j, k, -k; it satisfies g^2 = -1 for all g except +-1, and ijk = 1. (Note that the notation "-i" means "the product of i with -1" and -1 is understood to be central in Q.) Prove that every subgroup of Q is normal.<br />
<br />
6. (finishing example done in class) Let Gamma be the group F<x,y> / (x^2 = 1, y^2 = 1). (Remember, this means the quotient of the free group by the group generated by all conjugates of x^2 and y^2.) Prove that Gamma is infinite. Prove that Gamma has a subgroup R generated by xy isomorphic to Z, that R is normal, and that the quotient Gamma/R is isomorphic to Z/2Z. <br />
<br />
7. Suppose a group G has the property that every element in g satisfies g^2 = 1. Show that G must be abelian.<br />
<br />
==HOMEWORK 3 (due Oct 4)==<br />
<br />
1. Multiplication by -1 is an automorphism of Z/nZ. Let a be the homomorphism Z/2Z -> Aut(Z/nZ) which sends the nontrivial element of Z/2Z to multiplication by -1. Then we have a semidirect product of Z/pZ by Z/2Z as defined in class. This group of order 2n is called a ''dihedral group.'' and is denoted D_n (or sometimes D_{2n}).<br />
<br />
1a. Compute the center of D_n. (Note that the answer depends on n!)<br />
1b. Show that all involutions of D_n are conjugate to each other if and only if n is odd.<br />
<br />
2. (More on direct products that we didn't do in class.) Suppose that G is a semidirect product of N with H. Suppose furthermore that H is also normal in G. Prove that G is a direct product of N with H.<br />
<br />
3. Let Q be the quaternion group from last week's homework. Show that Q does not decompose as a semidirect product N \rtimes H, apart form the trivial cases N = {e} and N = Q.<br />
<br />
4. The ''affine linear group'' of degree n is the group of transformations from R^n to R^n of the form x -> Ax + b, where A is a matrix in GL_n(R) and b is a vector in R^n. Prove that the affine linear group is a semidirect product R^n \rtimes GL_n.<br />
<br />
5. The ''ordinary triangle group'' T(p,q,r) is the group with presentation <x,y | x^p = y^q = (xy)^r = 1>. This is a very interesting family of groups which arises naturally in many contexts in number theory and topology.<br />
<br />
6a. Prove that T(2,2,n) is isomorphic to the dihedral group D_n.<br />
6b. Prove that T(2,3,3) is finite and compute its order. Is it isomorphic to a finite group we've discussed in class?<br />
<br />
(It is a beautiful fact, but outside the scope of this course, that T(p,q,r) is finite if and only if 1/p + 1/q + 1/r > 1.)<br />
<br />
7. Let G be a subgroup of GL_n(R) which contains SL_n(R). Prove that G is a normal subgroup of GL_n(R).<br />
<br />
8. Let G be a group and let H_1 and H_2 be subgroups of G of finite index (that is, the number of cosets in G/H_i is finite, i = 1,2.) Prove that H_1 intersect H_2 also has finite index in G.<br />
<br />
9. Recall that the commutator [x,y] is x y x^{-1} y^{-1}. If G is a group, the commutator subgroup of G, denoted G' or [G,G], is the subgroup of G generated by all commutators [x,y] for x,y in G.<br />
<br />
9a. Show that G' is a normal subgroup of G.<br />
9b. Show that G/G' is an abelian group.<br />
9c. Show that if f: G -> A is a homomorphism, with A abelian, then G' is contained in ker f. Conclude that f factors through a homomorphism G/G' -> A.<br />
9d. Show that G' = G if and only if G has no nontrivial abelian quotient. In this case, we say G is ''perfect''.<br />
<br />
10. Under what conditions on (p,q,r) is the triangle group T(p,q,r) perfect?<br />
<br />
==HOMEWORK 4 (due Oct 16)==<br />
<br />
1. Let X be a set with a transitive action of a group G, which we think of as a homomorphism f: G -> Sym(X). Let H_x be the stabilizer of an element x of X.<br />
<br />
1a. If x' is another element of X, show that H_{x"} and H_x are conjugate subgroups. (Hint: this is not true without the hypothesis of transitivity, so make sure to use it!)<br />
<br />
1b. Show that the kernel of f is the intersection of all the conjugates of H_x in G. (This subgroup is called the ''normal core'' of H_x.)<br />
<br />
2. This week, a certified mathematical grown-up asked on MathOverflow what the index-3 subgroups of SL_2(Z/pZ) were. Let's do it ourselves for homework. Let p be a prime, and suppose H is an index-3 subgroup of SL_2(Z/pZ). I do not claim that the proof sketched below is necessarily the easiest, but it allows us to cover some ground that we won't get to do in lecture.<br />
<br />
2a. Using the coset action, show that the normal core of H is the kernel of a homomorphism f from SL_2(Z/pZ) to S_3.<br />
<br />
2b. A ''unipotent matrix'' is a matrix A such that (A-I)^n = 0 for some positive integer n. Prove that a unipotent element of SL_2(Z/pZ) has order dividing p.<br />
<br />
2c. You may use the fact that unipotent matrices generate SL_2(Z/pZ). Given this fact, show that f must be trivial when p > 3.<br />
<br />
2d. Using the above, show that SL_2(Z/pZ) has no index-3 subgroups when p > 3.<br />
<br />
2e. (extra credit) What can you say about the minimal index of a proper subgroup of SL_2(Z/pZ)?<br />
<br />
3. Let H be a subgroup of G of index 2. Prove that H is normal.<br />
<br />
4. We proved two theorems in class: Lagrange's theorem, which says that the order of a subgroup of G divides |G|, and Cauchy's theorem, which says that if p divides |G| then there exists a subgroup of G of order p. Cauchy's theorem is a kind of partial converse to Lagrange's theorem, but the full converse is false. Give an example of a finite group G and a divisor n of |G| such that you can prove there is no subgroup of G of order n.<br />
<br />
5. Let X be the set of lines in F_3^2 (here a line means a 1-dimensional linear subspace) so that |X| = 4. Let G be the projective linear group PGL_2(F_3), so that G acts on X via its action on F_3^2. This action can be thought of as a homomorphism from G to S_4. Show that this homomorphism is an ISOMORPHISM from PGL_2(F_3) to S_4.<br />
<br />
6. The group S_3 has only three conjugacy classes. Prove that a finite group with at most three conjugacy classes has order at most 6.<br />
<br />
7. Let X be the set of ordered triples of elements of {1,..,n}, for some n >= 3. Then S_n acts on X. How many orbits are there? (Hint: the answer does not depend on n.)<br />
<br />
8. Suppose that G and H are groups and f: G -> H is a homomorphism. Recall the definition of the abelianization G^ab from the previous problem set.<br />
<br />
8a. Show that the composition G -> H -> H^ab factors through a unique homomorphism G^ab -> H^ab, which we denote f^ab.<br />
<br />
8b. Show that if f: G -> H and g: H -> Q are homomorphisms, then f^ab o g^ab = (f o g)^ab.<br />
<br />
(For those reading MacLane, this constitutes a proof that abelianization is a '''functor''' from the category of groups to the category of abelian groups.)<br />
<br />
==HOMEWORK 5 (due Oct 23)==<br />
<br />
1. Show that there are only two different isomorphism classes of groups of order 14. (Hint: prove that such a group must be a semidirect product of a group of order 7 by a group of order 2.)<br />
<br />
2. (variant of Burnside's Lemma) Let G be a finite group and H a normal subgroup. Suppose G acts on a finite set X. <br />
<br />
2a. Show that if x and y lie in the same H-orbit, and g is an element of G, then gx and gy also lie in the same H-orbit. Thus, G/H has a natural action on the set of H-orbits X/H.<br />
<br />
2b. Show that the average number of fixed points in X of an element of the coset gH is the same as the number of fixed points of the element gH of G/H in its action on X/H.<br />
<br />
3. A ''central extension'' of a group G by an abelian group A is a group E, together with a surjective homomorphism E -> G whose kernel is central in E and is isomorphic to A.<br />
<br />
3a. Show that the quaternion group is a central extension of (Z/2Z)^2 by Z/2Z.<br />
<br />
3b. Give an example of another nonabelian group which is a central extension of (Z/2Z)^2 by Z/2Z and which is not isomorphic to the quaternion group.<br />
<br />
4. Prove that a central extension of an abelian group is nilpotent.<br />
<br />
5. Give two different composition series for S_4 and show that they have the same composition factors.<br />
<br />
6. Suppose that A is a finitely generated abelian group, and B is a subgroup of A.<br />
Prove that rank(A/B) = rank(A) - rank(B).<br />
ADDED: I asserted in class, but did not prove, that a subgroup of a finitely generated abelian group is again finitely generated. As one class member pointed out, without this fact it's not even clear why it makes sense to talk about rank(B)! You may assume it (though it would have been better for me to have included proving this fact as part of the problem....)<br />
<br />
7. For this problem, you may use the fact (which I should have asked on homework earlier) that if H_1 and H_2 are subgroups of G, then<br />
<br />
[G: H_1 intersect H_2] <= [G:H_1][G:H_2].<br />
<br />
Prove that if G is a simple group, d is an integer greater than 1, and S is a subgroup of index d in G, then |G| <= d^d.<br />
<br />
Conclude that, in particular, A_5 has no subgroup of index 3. On the other hand, exhibit a subgroup of A_5 of index 5.<br />
<br />
(Note: d^d is not a sharp bound; can you get a bound of d factorial instead?)<br />
<br />
<!--(This argument shows that a solvable subgroup of A_n has size at most about (n-1)!, but this is far from sharp; e.g. I found a citation to a 1967 paper of Dixon which shows that a solvable subgroup of A_n has size at most k^n where k is around 2.88.)--><br />
<br />
8. Give an example of an abelian group A with a subgroup B such that B is not a direct summand of A.<br />
<br />
9. Let A be the subgroup of Z^2 generated by the vector (1,0). Let B be the subgroup of Z^2 generated by the vector (3,3). Show that A and B are isomorphic to each other, but Z^2 / A is not isomorphic to Z^2 / B.<br />
<br />
==HOMEWORK 6 (due Oct 30)==<br />
<br />
1. In this problem, you will construct a functor F from the category of finite sets to the category of complex vector spaces. I'll tell you what it does on objects: if X is a finite set, then F(X) is the vector space spanned by a set of basis elements {e_x}_{x in X}, so that dim F(X) = |X|. Your job: complete the definition of F by describing the map from Mor_{Finite Sets}(X,Y) to Mor_{Vector Spaces}(F(X),F(Y)), and showing that your map respects composition.<br />
<br />
2. If (V, rho) is a representation of a group G, denote by V^G the space of vectors in V such that gv = v for all g in G; these are called _invariant_ vectors. We saw that the permutation representation of S_3 has a one-dimensional space of invariants. Prove that if X is a finite set with G-action, and V_X the corresponding permutation representation of G, then<br />
<br />
dim V_X^G = number of orbits of X.<br />
<br />
3. (Invariants are functorial) Let G be a group. Then there is a category called C[G]-Mod whose objects are complex representations of G (i.e. pairs (V,rho) where V is a complex vector space and rho is a homomorphism from G to GL(V)). The morphisms in this category are the ones described in class: Mor((V,rho),(W,psi)) is the set of linear maps f from V to W such that<br />
<br />
f(rho(g)(v)) = psi(g)(f(v))<br />
<br />
for all g in G and all v in V.<br />
<br />
Show that there is a functor H_0 from the category of representations of G to the category of vector spaces whose action on objects is given by<br />
<br />
H_0((V) = V^G.<br />
<br />
(In other words: your job again is to explain how to associate to a morphism from (V,rho) to (W,phi) a morphism from V^G to W^G, in a way that's compatible with composition.)<br />
<br />
<!--4. The group PSL_2(Z) has a presentation of the form [U,V|U^2 = V^3 = 1]. Thus, an n-dimensional representation of PSL_2(Z) is precisely a pair of n x n matrices (rho(U),rho(V)), such that rho(U)^2 = rho(V)^3 = 1. Prove that for every n > 1 there are uncountably many isomorphism classes of n-dimensional representations of PSL_2(Z). (Extra credit: show that there are uncountably many isomorphism classes of _irreducible_ representations.)<br />
<br />
An amazing theorem of Margulis (superrigidity) shows that SL_2(Z) is very special among arithmetic groups in this way. For instance, there are only _finitely_ many isomorphism classes of n-dimensional representations of PSL_d(Z) for any d > 2; those groups are "rigid."--><br />
<br />
5. Let G be a finite group, let X be a set with G-action, and let chi be the character of the corresponding permutation representation. Show that, for each g in G, chi(g) is the number of elements of X fixed by g. (I am actually going to prove this in class.)<br />
<br />
6. Let V be the space of homogeneous degree-3 polynomials in variables x_1, x_2, x_3; then S_3 acts on V by permutation of the three variables. Describe the decomposition of V into irreducible representations of S_3.<br />
<br />
7a. Show that if chi is the character of a representation of a finite group G, then chi(g^{-1}) is the complex conjugate of chi.<br />
<br />
7b. As a consequence, show that if g is conjugate to g^{-1} for all g in G, then chi(g) is real.<br />
<br />
7c. The groups D_p and S_n have the property that g is conjugate to g^{-1} for _every_ g in G, so that every character is a real-valued function on G. Give an example of a group G where this is not the case, and give an example of a representation of G whose character takes non-real values.<br />
<br />
==HOMEWORK 7 (due Nov 6)==<br />
<br />
1. Let G be a finite group, let X be a finite set with G-action, and let chi be the character of the corresponding permutation representation. Show that, for each g in G, chi(g) is the number of elements of X fixed by g. (I am actually going to prove this in class.)<br />
<br />
2a. Show that if chi is the character of a finite-dimensional representation of a finite group G, then chi(g^{-1}) is the complex conjugate of chi(g) for all g.<br />
<br />
2b. As a consequence, show that if g is conjugate to g^{-1} for all g in G, then chi(g) is real.<br />
<br />
2c. The groups D_p and S_n have the property that g is conjugate to g^{-1} for _every_ g in G, so that every character is a real-valued function on G. Give an example of a group G where this is not the case, and give an example of a representation of G whose character takes non-real values.<br />
<br />
3. Let V and W be complex vector spaces. Let f be a linear transformation of V, and g a linear transformation of W. <br />
<br />
3a. Show that there is a unique linear transformation F satisfying<br />
<br />
F(v tensor w) = f(v) tensor g(w)<br />
<br />
for all v in V and all w in W. We denote this transformation by f tensor g.<br />
<br />
3b. Suppose V and W are finite dimensional. Show that Trace(f tensor g) = Trace(f) Trace(g). What does this say when f and g are both the identity transformation?<br />
<br />
4. Let V be a vector space. We define Sym^2 V to be the quotient of V tensor V by the subspace generated by all elements of the form<br />
<br />
v tensor w - w tensor v<br />
<br />
for v,w in V.<br />
<br />
Suppose dim V = n. What is dim Sym^2 V?<br />
<br />
5. Show that Z/pZ tensor Z/qZ is zero when p and q are distinct primes.<br />
<br />
==HOMEWORK 8 (due Nov 13)==<br />
<br />
1. (Schur's lemma can be false over non-algebraically closed fields) Let G be Z/3Z and let g be a generator of G. Then G acts by cyclic permutations on the set {1,2,3}; let P be the corresponding permutation representation on the REAL (not complex!) vector space generated by e_1,e_2,e_3.<br />
<br />
1a. Show that P breaks up as the direct sum of the trivial representation and a 2-dimensional representation V which is irreducible. (Remember, we are not working over the complex numbers, so you cannot use the criterion for irreducibility in terms of the character that we worked out in class for the complex case; you need to prove directly that V has no nontrivial proper subrepresentation.)<br />
<br />
1b. Show that the map from V to V induced by g is G-equivariant, but is neither zero nor a scalar multiplication.<br />
<br />
2. If V_1 and V_2 are representations (not necessarily irreducible!) of G, show that <chi_V_1, chi V_2> is a non-negative integer.<br />
<br />
3. When a formula always yields a non-negative integer, it is often because it is secretly a formula for either the cardinality of some finite set or the dimension of some finite-dimensional vector space. In this case, show that<br />
<br />
<chi_{V_1}, chi_{V_2}> = dim_C Hom_G(V_1,V_2)<br />
<br />
where we recall that Hom_G(V_1,G_2) refers to the space of G-equivariant homomorphisms from V_1 to V_2.<br />
<br />
4. If A is an abelian group, show that every irreducible representation of A is 1-dimensional. (Hint: show that there is a vector which is an eigenvector for every element of A!)<br />
<br />
5. Let G be the group of order 20 which is the semidirect product of A by H, where A is Z/5Z and H is (Z/5Z)^* with its natural action on A. We will work out all the irreducible representations of A. <br />
<br />
5a. First of all, consider the 5-dimensional permutation representation of G acting by left multiplication on the cosets of H. Show that this representation decomposes as a direct sum of the trivial representation and an irreducible 4-dimensional representation. (Hint: it is probably easier to use the character of this representation than to prove irreducibility directly.) <br />
<br />
5b. Now show that there are 4 1-dimensional representations of G which factor through the quotient G/A (i.e. such that rho(A) is the identity.)<br />
<br />
5c. Using the decomposition of the regular representation, show that the irreducible representations of G you have constructed are the only ones.<br />
<br />
==HOMEWORK 9 (due Nov 20)==<br />
<br />
0. Find all ring homomorphisms from Z to Z/30Z. [Warning: there are 8 of them, not 30.]<br />
<br />
<br />
1. If R is a ring (not necessarily with 1) such that a^2 = a for every a in R, R is called a Boolean ring.<br />
<br />
1a. Show that any nonzero Boolean ring has characteristic 2. (In other words, show that a+a=0 for all a in R.)<br />
<br />
1b. Show that every Boolean ring is commutative.<br />
<br />
<br />
2. Let phi : R -> S be a homomorphism of commutative rings with 1.<br />
<br />
2a. If P is a prime ideal of S, show that phi^(-1) (P) is either R or a prime ideal of R.<br />
<br />
2b. If M is a maximal ideal of S and phi is surjective, show that phi^(-1) (M) is a maximal ideal of R.<br />
<br />
2c. Find a homomorphism phi : R -> S and a maximal ideal M of S such that phi^(-1) (M) is not a maximal ideal of R.<br />
<br />
<br />
3. Let Nil(A) denote the set of nilpotent elements of a ring A, and let R be a commutative ring. In class it was observed that Nil(R) is an ideal of R.<br />
<br />
3a. Find (a generator of) Nil(Z/720Z).<br />
<br />
3b. Show that Nil(R/Nil(R)) = 0.<br />
<br />
3c. If x is in Nil(R) and u is a unit, show that u+x is a unit.<br />
<br />
3d. Show that Nil(S), where S is the 2x2 matrices with real coefficients, is not an ideal of S.<br />
<br />
<br />
4a. If M is an R-module and I is an ideal of R such that im = 0 for all i in I and m in M, show that M carries the structure of an R/I-module.<br />
<br />
4b. If S is a commutative ring with 1 and J is a maximal ideal of S, show that J/J^2 is a vector space over S/J.<br />
<br />
<br />
5. A "divisible element" x of a Z-module is one such that for each positive integer k, there exists a y such that ky=x. Show that a free Z-module cannot contain a nonzero divisible element.<br />
<br />
<br />
6. Prove that the direct sum of any collection of free R-modules is also a free R-module.<br />
<br />
<br />
7. In this problem you will prove that the module M given by the direct product of copies of Z, indexed by the positive integers, is not a free Z-module. Therefore, assume that M is a free Z-module with basis B, and let N be the direct sum of copies of Z, as a submodule of M.<br />
<br />
7a. Show there is a countable subset B1 of B such that N is contained in the submodule N1 generated by B1. [Hint: N is countable.]<br />
<br />
7b. Show that M/N1 is a free Z-module.<br />
<br />
7c. Let S = {(b1, b2, b3, ...) : b_i = i! or -i!}. Show that there is some s in S which is not in N1. [Hint: S is uncountable.]<br />
<br />
7d. For any s in S but not N1, show that for each positive integer k, there exists some m in M with s-km in N1. Conclude that s+N1 is a divisible element of M/N1.<br />
<br />
7e. Use problem 5 to obtain a contradiction.<br />
<br />
<br />
8. In class I said "rank" is not always well-defined for R-modules if R is not commutative. Here is an example demonstrating this assertion. Let M be the countably infinite direct product of copies of Z as in exercise 7, and let R be the endomorphism ring of M. Define phi_1(a1,a2,a3,a4,...) = (a1,a3,a5,...) and phi_2(a1,a2,a3,a4,...) = (a2,a4,a6,...).<br />
<br />
8a. Show that {phi_1, phi_2} is a free basis for R, as a left R-module. [Hint: let psi_1(a1,a2,a3,...) = (a1,0,a2,0,...) and psi_2(a1,a2,a3,...) = (0,a1,0,a2,...). Compute the products of the phi_i and psi_j in either order, and use the result to show that the phi_i are a basis.]<br />
<br />
8b. Show that R is isomorphic to R^n for every positive integer n.</div>Dummithttps://hilbert.math.wisc.edu/wiki/index.php?title=741&diff=47197412012-11-19T03:45:56Z<p>Dummit: </p>
<hr />
<div>'''Math 741'''<br />
<br />
Algebra<br />
<br />
Prof: [http://www.math.wisc.edu/~ellenber Jordan Ellenberg]<br />
<br />
Grader: Evan Dummit<br />
<br />
Ellenberg's office hours: Friday 3pm.<br />
<br />
Grader's office hours: Monday 4pm. Late homework may be given directly to the grader, along with either (i) the instructor's permission, or (ii) a polite request for mercy.<br />
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This course, the first semester of the introductory graduate sequence in algebra, will cover the basic theory of groups, group actions, representation, linear and multilinear algebra, and the beginnings of ring theory.<br />
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==SYLLABUS==<br />
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In this space we will record the theorems and definitions we covered each week, which we can use as a list of notions you should be prepared to answer questions about on the Algebra qualifying exam. The material covered on the homework is also an excellent guide to the scope of the course. <br />
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'''WEEK 1''': <br />
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Definition of group. Associativity. Inverse. <br />
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Examples of group: GL_n(R). GL_n(Z). Z/nZ. R. Z. R^*. The free group F_k on k generators. <br />
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Homomorphisms. The homomorphisms from F_k to G are in bijection with G^k. Isomorphisms.<br />
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'''WEEK 2''':<br />
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The symmetric group (or permutation group) S_n on n letters. Cycle decomposition of a permutation. Order of a permutation. Thm: every element of a finite group has finite order. <br />
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Subgroups. Left and right cosets. Lagrange's Theorem. Cyclic groups. The order of an element of a finite group is a divisor of the order of the group.<br />
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The sign homomorphism S_n -> +-1.<br />
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'''WEEK 3'''<br />
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Normal subgroups. The quotient of a group by a normal subgroup. The first isomorphism theorem. Examples of S_n -> +-1 and S_4 -> S_3 with kernel V_4, the Klein 4-group.<br />
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Centralizers and centers. Abelian groups. The center of SL_n(R) is either 1 or +-1.<br />
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Groups with presentations. The infinite dihedral group <x,y | x^2 = 1, y^2 = 1>.<br />
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'''WEEK 4'''<br />
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More on groups with presentations.<br />
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Second and third isomorphism theorems.<br />
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Semidirect products.<br />
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'''WEEK 5'''<br />
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Group actions, orbits, and stabilizers.<br />
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Orbit-stabilizer theorem.<br />
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Cayley's theorem.<br />
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Cauchy's theorem.<br />
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'''WEEK 6'''<br />
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Applications of orbit-stabilizer theorem (p-groups have nontrivial center, first Sylow theorem.)<br />
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Classification of finite abelian groups and finitely generated abelian groups.<br />
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Composition series and the Jordan-Holder theorem (which we state but don't prove.) <br />
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The difference between knowing the composition factors and knowing the group (e.g. all p-groups of the same order have the same composition factors.)<br />
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'''WEEK 7'''<br />
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Simplicity of A_n.<br />
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Nilpotent groups (main example: the Heisenberg group)<br />
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Derived series and lower central series.<br />
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Category theory 101: Definition of category and functor. Some examples. A group is a groupoid with one object.<br />
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'''WEEK 8'''<br />
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Introduction to representation theory.<br />
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'''WEEK 10'''<br />
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Ring theory 101: Rings, ring homomorphisms, ideals, isomorphism theorems. Examples: fields, Z, the Hamilton quaternions, matrix rings, rings of polynomials and formal power series, quadratic integer rings, group rings. Integral domains. Maximal and prime ideals. The nilradical.<br />
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Module theory 101: Modules, module homomorphisms, submodules, isomorphism theorems. Noetherian modules and Zorn's lemma. Direct sums and direct products of arbitrary collections of modules.<br />
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Below you will find a repository of homework problems. Note that some of these problems are taken from Lang's ''Algebra''.<br />
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==HOMEWORK 1 (due Sep 20)==<br />
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1. Suppose that H_1 and H_2 are subgroups of a group G. Prove that the intersection of H_1 and H_2 is a subgroup of G.<br />
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2. Recall that S_3 (the symmetric group) is the group of permutations of the set {1..3}. List all the subgroups of S_3.<br />
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3. We can define an equivalence relation on rational numbers by declaring two rational numbers to be equal whenever they differ by an integer. We denote the set of equivalence classes by Q/Z. The operation of addition makes Q/Z into a group.<br />
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a) For each n, prove that Q/Z has a subgroup of order n.<br />
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b) Prove that Q/Z is a ''divisible'' group: that is, if x is an element of Q/Z and n is an integer, there exists an element y of Q/Z such that ny = x. (Note that we write the operation in this group as addition rather than multiplication, which is why we write ny for the n-fold product of y with itself rather than y^n)<br />
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c) Prove that Q/Z is not finitely generated. (Hint: prove that if x_1, .. x_d is a finite subset of Q/Z, the subgroup of Q/Z generated by x_1, ... x_d is finite.)<br />
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4. Let F_2 be the free group on two generators x,y. Prove that there is a automorphism of F_2 which sends x to xyx and y to xy, and prove that this automorphism is unique.<br />
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5. Let H be a subgroup of G, and let N_G(H), the ''normalizer'' of H in G, be the set of elements g in G satsifying g H g^{-1} = H. Prove that N_G(H) is a subgroup of G.<br />
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6. Let T be the subgroup of GL_2(R) consisting of diagonal matrices. This is an example of a "Cartan subgroup," or "torus" (whence the notation T.) Describe the normalizer N(T) of T in GL_2(R). Prove that there is a homomorphism from N(T) to Z/2Z whose kernel is precisely T. <br />
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7. Prove that the group of inner automorphisms of a group is normal in the full automorphism group Aut(G).<br />
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8. Let H and H' be subgroups of a group G, and let x be an element of G. We denote by HxH' the set of elements of the form hxh', with h in H and h' in H', and we call HxH a ''double coset'' of the pair (H,H').<br />
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a) Show that G decomposes as a disjoint union of double cosets of (H,H').<br />
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b) Let G be GL_2(R) and let B be the subgroup of upper triangular matrices. Prove that G decomposes into exactly two double cosets of (B,B).<br />
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c) Let G be the symmetric group S_n, and let H be the subgroup of permutations which fix 1. Describe the double coset decomposition of S_n into double cosets of (H,H).<br />
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==HOMEWORK 2 (due Sep 27)==<br />
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1. Prove that the symmetric group S_n is generated by transpositions. Hint: proceed by induction on n. Let H be the subgroup of S_n consisting of permutations fixing 1. Observe that H is isomorphic to S_{n-1}, so that H is generated by transpositions. Explain why it suffices to show that, for any g in S_n, the coset gH contains a transposition. Then explain why this is the case.<br />
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2. Given that the symmetric group S_n is generated by transpositions, show that there are no nontrivial homomorphisms from S_n to Z/pZ for any odd prime p.<br />
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3. Compute the centralizer of the triple flip (12)(34)(56) in S_6. (Note: this is the kind of problem that a computer algebra package like sage is very good at, and I encourage you to learn to use sage well enough to check your answer.)<br />
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4. Warning: a normal subgroup of a normal subgroup need not be normal! Let H be the subgroup of S_4 generated by (12)(34). Then H is a subgroup of the Klein 4-group V_4, which we have already shown is normal in S_4. Show that H is normal in V_4, but H is not normal in S_4.<br />
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5. The '''quaternion group''' Q is a non-abelian finite group of order 8. Its elements are 1, -1, i, -i, j, -j, k, -k; it satisfies g^2 = -1 for all g except +-1, and ijk = 1. (Note that the notation "-i" means "the product of i with -1" and -1 is understood to be central in Q.) Prove that every subgroup of Q is normal.<br />
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6. (finishing example done in class) Let Gamma be the group F<x,y> / (x^2 = 1, y^2 = 1). (Remember, this means the quotient of the free group by the group generated by all conjugates of x^2 and y^2.) Prove that Gamma is infinite. Prove that Gamma has a subgroup R generated by xy isomorphic to Z, that R is normal, and that the quotient Gamma/R is isomorphic to Z/2Z. <br />
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7. Suppose a group G has the property that every element in g satisfies g^2 = 1. Show that G must be abelian.<br />
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==HOMEWORK 3 (due Oct 4)==<br />
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1. Multiplication by -1 is an automorphism of Z/nZ. Let a be the homomorphism Z/2Z -> Aut(Z/nZ) which sends the nontrivial element of Z/2Z to multiplication by -1. Then we have a semidirect product of Z/pZ by Z/2Z as defined in class. This group of order 2n is called a ''dihedral group.'' and is denoted D_n (or sometimes D_{2n}).<br />
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1a. Compute the center of D_n. (Note that the answer depends on n!)<br />
1b. Show that all involutions of D_n are conjugate to each other if and only if n is odd.<br />
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2. (More on direct products that we didn't do in class.) Suppose that G is a semidirect product of N with H. Suppose furthermore that H is also normal in G. Prove that G is a direct product of N with H.<br />
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3. Let Q be the quaternion group from last week's homework. Show that Q does not decompose as a semidirect product N \rtimes H, apart form the trivial cases N = {e} and N = Q.<br />
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4. The ''affine linear group'' of degree n is the group of transformations from R^n to R^n of the form x -> Ax + b, where A is a matrix in GL_n(R) and b is a vector in R^n. Prove that the affine linear group is a semidirect product R^n \rtimes GL_n.<br />
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5. The ''ordinary triangle group'' T(p,q,r) is the group with presentation <x,y | x^p = y^q = (xy)^r = 1>. This is a very interesting family of groups which arises naturally in many contexts in number theory and topology.<br />
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6a. Prove that T(2,2,n) is isomorphic to the dihedral group D_n.<br />
6b. Prove that T(2,3,3) is finite and compute its order. Is it isomorphic to a finite group we've discussed in class?<br />
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(It is a beautiful fact, but outside the scope of this course, that T(p,q,r) is finite if and only if 1/p + 1/q + 1/r > 1.)<br />
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7. Let G be a subgroup of GL_n(R) which contains SL_n(R). Prove that G is a normal subgroup of GL_n(R).<br />
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8. Let G be a group and let H_1 and H_2 be subgroups of G of finite index (that is, the number of cosets in G/H_i is finite, i = 1,2.) Prove that H_1 intersect H_2 also has finite index in G.<br />
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9. Recall that the commutator [x,y] is x y x^{-1} y^{-1}. If G is a group, the commutator subgroup of G, denoted G' or [G,G], is the subgroup of G generated by all commutators [x,y] for x,y in G.<br />
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9a. Show that G' is a normal subgroup of G.<br />
9b. Show that G/G' is an abelian group.<br />
9c. Show that if f: G -> A is a homomorphism, with A abelian, then G' is contained in ker f. Conclude that f factors through a homomorphism G/G' -> A.<br />
9d. Show that G' = G if and only if G has no nontrivial abelian quotient. In this case, we say G is ''perfect''.<br />
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10. Under what conditions on (p,q,r) is the triangle group T(p,q,r) perfect?<br />
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==HOMEWORK 4 (due Oct 16)==<br />
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1. Let X be a set with a transitive action of a group G, which we think of as a homomorphism f: G -> Sym(X). Let H_x be the stabilizer of an element x of X.<br />
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1a. If x' is another element of X, show that H_{x"} and H_x are conjugate subgroups. (Hint: this is not true without the hypothesis of transitivity, so make sure to use it!)<br />
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1b. Show that the kernel of f is the intersection of all the conjugates of H_x in G. (This subgroup is called the ''normal core'' of H_x.)<br />
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2. This week, a certified mathematical grown-up asked on MathOverflow what the index-3 subgroups of SL_2(Z/pZ) were. Let's do it ourselves for homework. Let p be a prime, and suppose H is an index-3 subgroup of SL_2(Z/pZ). I do not claim that the proof sketched below is necessarily the easiest, but it allows us to cover some ground that we won't get to do in lecture.<br />
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2a. Using the coset action, show that the normal core of H is the kernel of a homomorphism f from SL_2(Z/pZ) to S_3.<br />
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2b. A ''unipotent matrix'' is a matrix A such that (A-I)^n = 0 for some positive integer n. Prove that a unipotent element of SL_2(Z/pZ) has order dividing p.<br />
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2c. You may use the fact that unipotent matrices generate SL_2(Z/pZ). Given this fact, show that f must be trivial when p > 3.<br />
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2d. Using the above, show that SL_2(Z/pZ) has no index-3 subgroups when p > 3.<br />
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2e. (extra credit) What can you say about the minimal index of a proper subgroup of SL_2(Z/pZ)?<br />
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3. Let H be a subgroup of G of index 2. Prove that H is normal.<br />
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4. We proved two theorems in class: Lagrange's theorem, which says that the order of a subgroup of G divides |G|, and Cauchy's theorem, which says that if p divides |G| then there exists a subgroup of G of order p. Cauchy's theorem is a kind of partial converse to Lagrange's theorem, but the full converse is false. Give an example of a finite group G and a divisor n of |G| such that you can prove there is no subgroup of G of order n.<br />
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5. Let X be the set of lines in F_3^2 (here a line means a 1-dimensional linear subspace) so that |X| = 4. Let G be the projective linear group PGL_2(F_3), so that G acts on X via its action on F_3^2. This action can be thought of as a homomorphism from G to S_4. Show that this homomorphism is an ISOMORPHISM from PGL_2(F_3) to S_4.<br />
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6. The group S_3 has only three conjugacy classes. Prove that a finite group with at most three conjugacy classes has order at most 6.<br />
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7. Let X be the set of ordered triples of elements of {1,..,n}, for some n >= 3. Then S_n acts on X. How many orbits are there? (Hint: the answer does not depend on n.)<br />
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8. Suppose that G and H are groups and f: G -> H is a homomorphism. Recall the definition of the abelianization G^ab from the previous problem set.<br />
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8a. Show that the composition G -> H -> H^ab factors through a unique homomorphism G^ab -> H^ab, which we denote f^ab.<br />
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8b. Show that if f: G -> H and g: H -> Q are homomorphisms, then f^ab o g^ab = (f o g)^ab.<br />
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(For those reading MacLane, this constitutes a proof that abelianization is a '''functor''' from the category of groups to the category of abelian groups.)<br />
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==HOMEWORK 5 (due Oct 23)==<br />
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1. Show that there are only two different isomorphism classes of groups of order 14. (Hint: prove that such a group must be a semidirect product of a group of order 7 by a group of order 2.)<br />
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2. (variant of Burnside's Lemma) Let G be a finite group and H a normal subgroup. Suppose G acts on a finite set X. <br />
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2a. Show that if x and y lie in the same H-orbit, and g is an element of G, then gx and gy also lie in the same H-orbit. Thus, G/H has a natural action on the set of H-orbits X/H.<br />
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2b. Show that the average number of fixed points in X of an element of the coset gH is the same as the number of fixed points of the element gH of G/H in its action on X/H.<br />
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3. A ''central extension'' of a group G by an abelian group A is a group E, together with a surjective homomorphism E -> G whose kernel is central in E and is isomorphic to A.<br />
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3a. Show that the quaternion group is a central extension of (Z/2Z)^2 by Z/2Z.<br />
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3b. Give an example of another nonabelian group which is a central extension of (Z/2Z)^2 by Z/2Z and which is not isomorphic to the quaternion group.<br />
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4. Prove that a central extension of an abelian group is nilpotent.<br />
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5. Give two different composition series for S_4 and show that they have the same composition factors.<br />
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6. Suppose that A is a finitely generated abelian group, and B is a subgroup of A.<br />
Prove that rank(A/B) = rank(A) - rank(B).<br />
ADDED: I asserted in class, but did not prove, that a subgroup of a finitely generated abelian group is again finitely generated. As one class member pointed out, without this fact it's not even clear why it makes sense to talk about rank(B)! You may assume it (though it would have been better for me to have included proving this fact as part of the problem....)<br />
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7. For this problem, you may use the fact (which I should have asked on homework earlier) that if H_1 and H_2 are subgroups of G, then<br />
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[G: H_1 intersect H_2] <= [G:H_1][G:H_2].<br />
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Prove that if G is a simple group, d is an integer greater than 1, and S is a subgroup of index d in G, then |G| <= d^d.<br />
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Conclude that, in particular, A_5 has no subgroup of index 3. On the other hand, exhibit a subgroup of A_5 of index 5.<br />
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(Note: d^d is not a sharp bound; can you get a bound of d factorial instead?)<br />
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<!--(This argument shows that a solvable subgroup of A_n has size at most about (n-1)!, but this is far from sharp; e.g. I found a citation to a 1967 paper of Dixon which shows that a solvable subgroup of A_n has size at most k^n where k is around 2.88.)--><br />
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8. Give an example of an abelian group A with a subgroup B such that B is not a direct summand of A.<br />
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9. Let A be the subgroup of Z^2 generated by the vector (1,0). Let B be the subgroup of Z^2 generated by the vector (3,3). Show that A and B are isomorphic to each other, but Z^2 / A is not isomorphic to Z^2 / B.<br />
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==HOMEWORK 6 (due Oct 30)==<br />
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1. In this problem, you will construct a functor F from the category of finite sets to the category of complex vector spaces. I'll tell you what it does on objects: if X is a finite set, then F(X) is the vector space spanned by a set of basis elements {e_x}_{x in X}, so that dim F(X) = |X|. Your job: complete the definition of F by describing the map from Mor_{Finite Sets}(X,Y) to Mor_{Vector Spaces}(F(X),F(Y)), and showing that your map respects composition.<br />
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2. If (V, rho) is a representation of a group G, denote by V^G the space of vectors in V such that gv = v for all g in G; these are called _invariant_ vectors. We saw that the permutation representation of S_3 has a one-dimensional space of invariants. Prove that if X is a finite set with G-action, and V_X the corresponding permutation representation of G, then<br />
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dim V_X^G = number of orbits of X.<br />
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3. (Invariants are functorial) Let G be a group. Then there is a category called C[G]-Mod whose objects are complex representations of G (i.e. pairs (V,rho) where V is a complex vector space and rho is a homomorphism from G to GL(V)). The morphisms in this category are the ones described in class: Mor((V,rho),(W,psi)) is the set of linear maps f from V to W such that<br />
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f(rho(g)(v)) = psi(g)(f(v))<br />
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for all g in G and all v in V.<br />
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Show that there is a functor H_0 from the category of representations of G to the category of vector spaces whose action on objects is given by<br />
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H_0((V) = V^G.<br />
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(In other words: your job again is to explain how to associate to a morphism from (V,rho) to (W,phi) a morphism from V^G to W^G, in a way that's compatible with composition.)<br />
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<!--4. The group PSL_2(Z) has a presentation of the form [U,V|U^2 = V^3 = 1]. Thus, an n-dimensional representation of PSL_2(Z) is precisely a pair of n x n matrices (rho(U),rho(V)), such that rho(U)^2 = rho(V)^3 = 1. Prove that for every n > 1 there are uncountably many isomorphism classes of n-dimensional representations of PSL_2(Z). (Extra credit: show that there are uncountably many isomorphism classes of _irreducible_ representations.)<br />
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An amazing theorem of Margulis (superrigidity) shows that SL_2(Z) is very special among arithmetic groups in this way. For instance, there are only _finitely_ many isomorphism classes of n-dimensional representations of PSL_d(Z) for any d > 2; those groups are "rigid."--><br />
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5. Let G be a finite group, let X be a set with G-action, and let chi be the character of the corresponding permutation representation. Show that, for each g in G, chi(g) is the number of elements of X fixed by g. (I am actually going to prove this in class.)<br />
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6. Let V be the space of homogeneous degree-3 polynomials in variables x_1, x_2, x_3; then S_3 acts on V by permutation of the three variables. Describe the decomposition of V into irreducible representations of S_3.<br />
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7a. Show that if chi is the character of a representation of a finite group G, then chi(g^{-1}) is the complex conjugate of chi.<br />
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7b. As a consequence, show that if g is conjugate to g^{-1} for all g in G, then chi(g) is real.<br />
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7c. The groups D_p and S_n have the property that g is conjugate to g^{-1} for _every_ g in G, so that every character is a real-valued function on G. Give an example of a group G where this is not the case, and give an example of a representation of G whose character takes non-real values.<br />
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==HOMEWORK 7 (due Nov 6)==<br />
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1. Let G be a finite group, let X be a finite set with G-action, and let chi be the character of the corresponding permutation representation. Show that, for each g in G, chi(g) is the number of elements of X fixed by g. (I am actually going to prove this in class.)<br />
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2a. Show that if chi is the character of a finite-dimensional representation of a finite group G, then chi(g^{-1}) is the complex conjugate of chi(g) for all g.<br />
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2b. As a consequence, show that if g is conjugate to g^{-1} for all g in G, then chi(g) is real.<br />
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2c. The groups D_p and S_n have the property that g is conjugate to g^{-1} for _every_ g in G, so that every character is a real-valued function on G. Give an example of a group G where this is not the case, and give an example of a representation of G whose character takes non-real values.<br />
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3. Let V and W be complex vector spaces. Let f be a linear transformation of V, and g a linear transformation of W. <br />
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3a. Show that there is a unique linear transformation F satisfying<br />
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F(v tensor w) = f(v) tensor g(w)<br />
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for all v in V and all w in W. We denote this transformation by f tensor g.<br />
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3b. Suppose V and W are finite dimensional. Show that Trace(f tensor g) = Trace(f) Trace(g). What does this say when f and g are both the identity transformation?<br />
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4. Let V be a vector space. We define Sym^2 V to be the quotient of V tensor V by the subspace generated by all elements of the form<br />
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v tensor w - w tensor v<br />
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for v,w in V.<br />
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Suppose dim V = n. What is dim Sym^2 V?<br />
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5. Show that Z/pZ tensor Z/qZ is zero when p and q are distinct primes.<br />
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==HOMEWORK 8 (due Nov 13)==<br />
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1. (Schur's lemma can be false over non-algebraically closed fields) Let G be Z/3Z and let g be a generator of G. Then G acts by cyclic permutations on the set {1,2,3}; let P be the corresponding permutation representation on the REAL (not complex!) vector space generated by e_1,e_2,e_3.<br />
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1a. Show that P breaks up as the direct sum of the trivial representation and a 2-dimensional representation V which is irreducible. (Remember, we are not working over the complex numbers, so you cannot use the criterion for irreducibility in terms of the character that we worked out in class for the complex case; you need to prove directly that V has no nontrivial proper subrepresentation.)<br />
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1b. Show that the map from V to V induced by g is G-equivariant, but is neither zero nor a scalar multiplication.<br />
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2. If V_1 and V_2 are representations (not necessarily irreducible!) of G, show that <chi_V_1, chi V_2> is a non-negative integer.<br />
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3. When a formula always yields a non-negative integer, it is often because it is secretly a formula for either the cardinality of some finite set or the dimension of some finite-dimensional vector space. In this case, show that<br />
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<chi_{V_1}, chi_{V_2}> = dim_C Hom_G(V_1,V_2)<br />
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where we recall that Hom_G(V_1,G_2) refers to the space of G-equivariant homomorphisms from V_1 to V_2.<br />
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4. If A is an abelian group, show that every irreducible representation of A is 1-dimensional. (Hint: show that there is a vector which is an eigenvector for every element of A!)<br />
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5. Let G be the group of order 20 which is the semidirect product of A by H, where A is Z/5Z and H is (Z/5Z)^* with its natural action on A. We will work out all the irreducible representations of A. <br />
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5a. First of all, consider the 5-dimensional permutation representation of G acting by left multiplication on the cosets of H. Show that this representation decomposes as a direct sum of the trivial representation and an irreducible 4-dimensional representation. (Hint: it is probably easier to use the character of this representation than to prove irreducibility directly.) <br />
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5b. Now show that there are 4 1-dimensional representations of G which factor through the quotient G/A (i.e. such that rho(A) is the identity.)<br />
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5c. Using the decomposition of the regular representation, show that the irreducible representations of G you have constructed are the only ones.<br />
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==HOMEWORK 9 (due Nov 20)==<br />
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1. If R is a ring (not necessarily with 1) such that a^2 = a for every a in R, R is called a Boolean ring.<br />
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1a. Show that every Boolean ring is commutative.<br />
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1b. Show that any nonzero Boolean ring has characteristic 2. (In other words, show that a+a=0 for all a in R.)<br />
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2. Let phi : R -> S be a homomorphism of commutative rings with 1.<br />
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2a. If P is a prime ideal of S, show that phi^(-1) (P) is either R or a prime ideal of R.<br />
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2b. If M is a maximal ideal of S and phi is surjective, show that phi^(-1) (M) is a maximal ideal of R.<br />
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2c. Find a homomorphism phi : R -> S and a maximal ideal M of S such that phi^(-1) (M) is not a maximal ideal of R.<br />
<br />
<br />
3. Let Nil(A) denote the set of nilpotent elements of a ring A, and let R be a commutative ring. In class it was observed that Nil(R) is an ideal of R.<br />
<br />
3a. Find (a generator of) Nil(Z/720Z).<br />
<br />
3b. Show that Nil(R/Nil(R)) = 0.<br />
<br />
3c. If x is in Nil(R) and u is a unit, show that u+x is a unit.<br />
<br />
3d. Show that Nil(S), where S is the 2x2 matrices with real coefficients, is not an ideal of S.<br />
<br />
<br />
4a. If M is an R-module and I is an ideal of R such that im = 0 for all i in I and m in M, show that M carries the structure of an R/I-module.<br />
<br />
4b. If S is a commutative ring with 1 and J is a maximal ideal of S, show that J/J^2 is a vector space over S/J.<br />
<br />
<br />
5. A "divisible element" x of a Z-module is one such that for each positive integer k, there exists a y such that ky=x. Show that a free Z-module cannot contain a nonzero divisible element.<br />
<br />
<br />
6. Prove that the direct sum of any collection of free R-modules is also a free R-module.<br />
<br />
<br />
7. In this problem you will prove that the module M given by the direct product of copies of Z, indexed by the positive integers, is not a free Z-module. Therefore, assume that M is a free Z-module with basis B, and let N be the direct sum of copies of Z, as a submodule of M.<br />
<br />
7a. Show there is a countable subset B1 of B such that N is contained in the submodule N1 generated by B1. [Hint: N is countable.]<br />
<br />
7b. Show that M/N1 is a free Z-module.<br />
<br />
7c. Let S = {(b1, b2, b3, ...) : b_i = i! or -i!}. Show that there is some s in S which is not in N1. [Hint: S is uncountable.]<br />
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7d. For any s in S but not N1, show that for each positive integer k, there exists some m in M with s-km in N1. Conclude that s+N1 is a divisible element of M/N1.<br />
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7e. Use problem 5 to obtain a contradiction.<br />
<br />
<br />
8. In class I said "rank" is not always well-defined for R-modules if R is not commutative. Here is an example demonstrating this assertion. Let M be the countably infinite direct product of copies of Z as in exercise 7, and let R be the endomorphism ring of M. Define phi_1(a1,a2,a3,a4,...) = (a1,a3,a5,...) and phi_2(a1,a2,a3,a4,...) = (a2,a4,a6,...).<br />
<br />
8a. Show that {phi_1, phi_2} is a free basis for R, as a left R-module. [Hint: let psi_1(a1,a2,a3,...) = (a1,0,a2,0,...) and psi_2(a1,a2,a3,...) = (0,a1,0,a2,...). Compute the products of the phi_i and psi_j in either order, and use the result to show that the phi_i are a basis.]<br />
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8b. Show that R is isomorphic to R^n for every positive integer n.</div>Dummithttps://hilbert.math.wisc.edu/wiki/index.php?title=741&diff=47187412012-11-19T01:19:19Z<p>Dummit: </p>
<hr />
<div>'''Math 741'''<br />
<br />
Algebra<br />
<br />
Prof: [http://www.math.wisc.edu/~ellenber Jordan Ellenberg]<br />
<br />
Grader: Evan Dummit<br />
<br />
Ellenberg's office hours: Friday 3pm.<br />
<br />
Grader's office hours: Monday 4pm. Late homework may be given directly to the grader, along with either (i) the instructor's permission, or (ii) a polite request for mercy.<br />
<br />
This course, the first semester of the introductory graduate sequence in algebra, will cover the basic theory of groups, group actions, representation, linear and multilinear algebra, and the beginnings of ring theory.<br />
<br />
==SYLLABUS==<br />
<br />
In this space we will record the theorems and definitions we covered each week, which we can use as a list of notions you should be prepared to answer questions about on the Algebra qualifying exam. The material covered on the homework is also an excellent guide to the scope of the course. <br />
<br />
'''WEEK 1''': <br />
<br />
Definition of group. Associativity. Inverse. <br />
<br />
Examples of group: GL_n(R). GL_n(Z). Z/nZ. R. Z. R^*. The free group F_k on k generators. <br />
<br />
Homomorphisms. The homomorphisms from F_k to G are in bijection with G^k. Isomorphisms.<br />
<br />
'''WEEK 2''':<br />
<br />
The symmetric group (or permutation group) S_n on n letters. Cycle decomposition of a permutation. Order of a permutation. Thm: every element of a finite group has finite order. <br />
<br />
Subgroups. Left and right cosets. Lagrange's Theorem. Cyclic groups. The order of an element of a finite group is a divisor of the order of the group.<br />
<br />
The sign homomorphism S_n -> +-1.<br />
<br />
<br />
'''WEEK 3'''<br />
<br />
Normal subgroups. The quotient of a group by a normal subgroup. The first isomorphism theorem. Examples of S_n -> +-1 and S_4 -> S_3 with kernel V_4, the Klein 4-group.<br />
<br />
Centralizers and centers. Abelian groups. The center of SL_n(R) is either 1 or +-1.<br />
<br />
Groups with presentations. The infinite dihedral group <x,y | x^2 = 1, y^2 = 1>.<br />
<br />
'''WEEK 4'''<br />
<br />
More on groups with presentations.<br />
<br />
Second and third isomorphism theorems.<br />
<br />
Semidirect products.<br />
<br />
'''WEEK 5'''<br />
<br />
Group actions, orbits, and stabilizers.<br />
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Orbit-stabilizer theorem.<br />
<br />
Cayley's theorem.<br />
<br />
Cauchy's theorem.<br />
<br />
'''WEEK 6'''<br />
<br />
Applications of orbit-stabilizer theorem (p-groups have nontrivial center, first Sylow theorem.)<br />
<br />
Classification of finite abelian groups and finitely generated abelian groups.<br />
<br />
Composition series and the Jordan-Holder theorem (which we state but don't prove.) <br />
<br />
The difference between knowing the composition factors and knowing the group (e.g. all p-groups of the same order have the same composition factors.)<br />
<br />
'''WEEK 7'''<br />
<br />
Simplicity of A_n.<br />
<br />
Nilpotent groups (main example: the Heisenberg group)<br />
<br />
Derived series and lower central series.<br />
<br />
Category theory 101: Definition of category and functor. Some examples. A group is a groupoid with one object.<br />
<br />
'''WEEK 8'''<br />
<br />
Introduction to representation theory. <br />
<br />
<br />
<br />
Below you will find a repository of homework problems. Note that some of these problems are taken from Lang's ''Algebra''.<br />
<br />
==HOMEWORK 1 (due Sep 20)==<br />
<br />
1. Suppose that H_1 and H_2 are subgroups of a group G. Prove that the intersection of H_1 and H_2 is a subgroup of G.<br />
<br />
2. Recall that S_3 (the symmetric group) is the group of permutations of the set {1..3}. List all the subgroups of S_3.<br />
<br />
3. We can define an equivalence relation on rational numbers by declaring two rational numbers to be equal whenever they differ by an integer. We denote the set of equivalence classes by Q/Z. The operation of addition makes Q/Z into a group.<br />
<br />
a) For each n, prove that Q/Z has a subgroup of order n.<br />
<br />
b) Prove that Q/Z is a ''divisible'' group: that is, if x is an element of Q/Z and n is an integer, there exists an element y of Q/Z such that ny = x. (Note that we write the operation in this group as addition rather than multiplication, which is why we write ny for the n-fold product of y with itself rather than y^n)<br />
<br />
c) Prove that Q/Z is not finitely generated. (Hint: prove that if x_1, .. x_d is a finite subset of Q/Z, the subgroup of Q/Z generated by x_1, ... x_d is finite.)<br />
<br />
4. Let F_2 be the free group on two generators x,y. Prove that there is a automorphism of F_2 which sends x to xyx and y to xy, and prove that this automorphism is unique.<br />
<br />
5. Let H be a subgroup of G, and let N_G(H), the ''normalizer'' of H in G, be the set of elements g in G satsifying g H g^{-1} = H. Prove that N_G(H) is a subgroup of G.<br />
<br />
6. Let T be the subgroup of GL_2(R) consisting of diagonal matrices. This is an example of a "Cartan subgroup," or "torus" (whence the notation T.) Describe the normalizer N(T) of T in GL_2(R). Prove that there is a homomorphism from N(T) to Z/2Z whose kernel is precisely T. <br />
<br />
7. Prove that the group of inner automorphisms of a group is normal in the full automorphism group Aut(G).<br />
<br />
8. Let H and H' be subgroups of a group G, and let x be an element of G. We denote by HxH' the set of elements of the form hxh', with h in H and h' in H', and we call HxH a ''double coset'' of the pair (H,H').<br />
<br />
a) Show that G decomposes as a disjoint union of double cosets of (H,H').<br />
<br />
b) Let G be GL_2(R) and let B be the subgroup of upper triangular matrices. Prove that G decomposes into exactly two double cosets of (B,B).<br />
<br />
c) Let G be the symmetric group S_n, and let H be the subgroup of permutations which fix 1. Describe the double coset decomposition of S_n into double cosets of (H,H).<br />
<br />
==HOMEWORK 2 (due Sep 27)==<br />
<br />
1. Prove that the symmetric group S_n is generated by transpositions. Hint: proceed by induction on n. Let H be the subgroup of S_n consisting of permutations fixing 1. Observe that H is isomorphic to S_{n-1}, so that H is generated by transpositions. Explain why it suffices to show that, for any g in S_n, the coset gH contains a transposition. Then explain why this is the case.<br />
<br />
2. Given that the symmetric group S_n is generated by transpositions, show that there are no nontrivial homomorphisms from S_n to Z/pZ for any odd prime p.<br />
<br />
3. Compute the centralizer of the triple flip (12)(34)(56) in S_6. (Note: this is the kind of problem that a computer algebra package like sage is very good at, and I encourage you to learn to use sage well enough to check your answer.)<br />
<br />
4. Warning: a normal subgroup of a normal subgroup need not be normal! Let H be the subgroup of S_4 generated by (12)(34). Then H is a subgroup of the Klein 4-group V_4, which we have already shown is normal in S_4. Show that H is normal in V_4, but H is not normal in S_4.<br />
<br />
5. The '''quaternion group''' Q is a non-abelian finite group of order 8. Its elements are 1, -1, i, -i, j, -j, k, -k; it satisfies g^2 = -1 for all g except +-1, and ijk = 1. (Note that the notation "-i" means "the product of i with -1" and -1 is understood to be central in Q.) Prove that every subgroup of Q is normal.<br />
<br />
6. (finishing example done in class) Let Gamma be the group F<x,y> / (x^2 = 1, y^2 = 1). (Remember, this means the quotient of the free group by the group generated by all conjugates of x^2 and y^2.) Prove that Gamma is infinite. Prove that Gamma has a subgroup R generated by xy isomorphic to Z, that R is normal, and that the quotient Gamma/R is isomorphic to Z/2Z. <br />
<br />
7. Suppose a group G has the property that every element in g satisfies g^2 = 1. Show that G must be abelian.<br />
<br />
==HOMEWORK 3 (due Oct 4)==<br />
<br />
1. Multiplication by -1 is an automorphism of Z/nZ. Let a be the homomorphism Z/2Z -> Aut(Z/nZ) which sends the nontrivial element of Z/2Z to multiplication by -1. Then we have a semidirect product of Z/pZ by Z/2Z as defined in class. This group of order 2n is called a ''dihedral group.'' and is denoted D_n (or sometimes D_{2n}).<br />
<br />
1a. Compute the center of D_n. (Note that the answer depends on n!)<br />
1b. Show that all involutions of D_n are conjugate to each other if and only if n is odd.<br />
<br />
2. (More on direct products that we didn't do in class.) Suppose that G is a semidirect product of N with H. Suppose furthermore that H is also normal in G. Prove that G is a direct product of N with H.<br />
<br />
3. Let Q be the quaternion group from last week's homework. Show that Q does not decompose as a semidirect product N \rtimes H, apart form the trivial cases N = {e} and N = Q.<br />
<br />
4. The ''affine linear group'' of degree n is the group of transformations from R^n to R^n of the form x -> Ax + b, where A is a matrix in GL_n(R) and b is a vector in R^n. Prove that the affine linear group is a semidirect product R^n \rtimes GL_n.<br />
<br />
5. The ''ordinary triangle group'' T(p,q,r) is the group with presentation <x,y | x^p = y^q = (xy)^r = 1>. This is a very interesting family of groups which arises naturally in many contexts in number theory and topology.<br />
<br />
6a. Prove that T(2,2,n) is isomorphic to the dihedral group D_n.<br />
6b. Prove that T(2,3,3) is finite and compute its order. Is it isomorphic to a finite group we've discussed in class?<br />
<br />
(It is a beautiful fact, but outside the scope of this course, that T(p,q,r) is finite if and only if 1/p + 1/q + 1/r > 1.)<br />
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7. Let G be a subgroup of GL_n(R) which contains SL_n(R). Prove that G is a normal subgroup of GL_n(R).<br />
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8. Let G be a group and let H_1 and H_2 be subgroups of G of finite index (that is, the number of cosets in G/H_i is finite, i = 1,2.) Prove that H_1 intersect H_2 also has finite index in G.<br />
<br />
9. Recall that the commutator [x,y] is x y x^{-1} y^{-1}. If G is a group, the commutator subgroup of G, denoted G' or [G,G], is the subgroup of G generated by all commutators [x,y] for x,y in G.<br />
<br />
9a. Show that G' is a normal subgroup of G.<br />
9b. Show that G/G' is an abelian group.<br />
9c. Show that if f: G -> A is a homomorphism, with A abelian, then G' is contained in ker f. Conclude that f factors through a homomorphism G/G' -> A.<br />
9d. Show that G' = G if and only if G has no nontrivial abelian quotient. In this case, we say G is ''perfect''.<br />
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10. Under what conditions on (p,q,r) is the triangle group T(p,q,r) perfect?<br />
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==HOMEWORK 4 (due Oct 16)==<br />
<br />
1. Let X be a set with a transitive action of a group G, which we think of as a homomorphism f: G -> Sym(X). Let H_x be the stabilizer of an element x of X.<br />
<br />
1a. If x' is another element of X, show that H_{x"} and H_x are conjugate subgroups. (Hint: this is not true without the hypothesis of transitivity, so make sure to use it!)<br />
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1b. Show that the kernel of f is the intersection of all the conjugates of H_x in G. (This subgroup is called the ''normal core'' of H_x.)<br />
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2. This week, a certified mathematical grown-up asked on MathOverflow what the index-3 subgroups of SL_2(Z/pZ) were. Let's do it ourselves for homework. Let p be a prime, and suppose H is an index-3 subgroup of SL_2(Z/pZ). I do not claim that the proof sketched below is necessarily the easiest, but it allows us to cover some ground that we won't get to do in lecture.<br />
<br />
2a. Using the coset action, show that the normal core of H is the kernel of a homomorphism f from SL_2(Z/pZ) to S_3.<br />
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2b. A ''unipotent matrix'' is a matrix A such that (A-I)^n = 0 for some positive integer n. Prove that a unipotent element of SL_2(Z/pZ) has order dividing p.<br />
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2c. You may use the fact that unipotent matrices generate SL_2(Z/pZ). Given this fact, show that f must be trivial when p > 3.<br />
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2d. Using the above, show that SL_2(Z/pZ) has no index-3 subgroups when p > 3.<br />
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2e. (extra credit) What can you say about the minimal index of a proper subgroup of SL_2(Z/pZ)?<br />
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3. Let H be a subgroup of G of index 2. Prove that H is normal.<br />
<br />
4. We proved two theorems in class: Lagrange's theorem, which says that the order of a subgroup of G divides |G|, and Cauchy's theorem, which says that if p divides |G| then there exists a subgroup of G of order p. Cauchy's theorem is a kind of partial converse to Lagrange's theorem, but the full converse is false. Give an example of a finite group G and a divisor n of |G| such that you can prove there is no subgroup of G of order n.<br />
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5. Let X be the set of lines in F_3^2 (here a line means a 1-dimensional linear subspace) so that |X| = 4. Let G be the projective linear group PGL_2(F_3), so that G acts on X via its action on F_3^2. This action can be thought of as a homomorphism from G to S_4. Show that this homomorphism is an ISOMORPHISM from PGL_2(F_3) to S_4.<br />
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6. The group S_3 has only three conjugacy classes. Prove that a finite group with at most three conjugacy classes has order at most 6.<br />
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7. Let X be the set of ordered triples of elements of {1,..,n}, for some n >= 3. Then S_n acts on X. How many orbits are there? (Hint: the answer does not depend on n.)<br />
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8. Suppose that G and H are groups and f: G -> H is a homomorphism. Recall the definition of the abelianization G^ab from the previous problem set.<br />
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8a. Show that the composition G -> H -> H^ab factors through a unique homomorphism G^ab -> H^ab, which we denote f^ab.<br />
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8b. Show that if f: G -> H and g: H -> Q are homomorphisms, then f^ab o g^ab = (f o g)^ab.<br />
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(For those reading MacLane, this constitutes a proof that abelianization is a '''functor''' from the category of groups to the category of abelian groups.)<br />
<br />
==HOMEWORK 5 (due Oct 23)==<br />
<br />
1. Show that there are only two different isomorphism classes of groups of order 14. (Hint: prove that such a group must be a semidirect product of a group of order 7 by a group of order 2.)<br />
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2. (variant of Burnside's Lemma) Let G be a finite group and H a normal subgroup. Suppose G acts on a finite set X. <br />
<br />
2a. Show that if x and y lie in the same H-orbit, and g is an element of G, then gx and gy also lie in the same H-orbit. Thus, G/H has a natural action on the set of H-orbits X/H.<br />
<br />
2b. Show that the average number of fixed points in X of an element of the coset gH is the same as the number of fixed points of the element gH of G/H in its action on X/H.<br />
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3. A ''central extension'' of a group G by an abelian group A is a group E, together with a surjective homomorphism E -> G whose kernel is central in E and is isomorphic to A.<br />
<br />
3a. Show that the quaternion group is a central extension of (Z/2Z)^2 by Z/2Z.<br />
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3b. Give an example of another nonabelian group which is a central extension of (Z/2Z)^2 by Z/2Z and which is not isomorphic to the quaternion group.<br />
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4. Prove that a central extension of an abelian group is nilpotent.<br />
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5. Give two different composition series for S_4 and show that they have the same composition factors.<br />
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6. Suppose that A is a finitely generated abelian group, and B is a subgroup of A.<br />
Prove that rank(A/B) = rank(A) - rank(B).<br />
ADDED: I asserted in class, but did not prove, that a subgroup of a finitely generated abelian group is again finitely generated. As one class member pointed out, without this fact it's not even clear why it makes sense to talk about rank(B)! You may assume it (though it would have been better for me to have included proving this fact as part of the problem....)<br />
<br />
7. For this problem, you may use the fact (which I should have asked on homework earlier) that if H_1 and H_2 are subgroups of G, then<br />
<br />
[G: H_1 intersect H_2] <= [G:H_1][G:H_2].<br />
<br />
Prove that if G is a simple group, d is an integer greater than 1, and S is a subgroup of index d in G, then |G| <= d^d.<br />
<br />
Conclude that, in particular, A_5 has no subgroup of index 3. On the other hand, exhibit a subgroup of A_5 of index 5.<br />
<br />
(Note: d^d is not a sharp bound; can you get a bound of d factorial instead?)<br />
<br />
<!--(This argument shows that a solvable subgroup of A_n has size at most about (n-1)!, but this is far from sharp; e.g. I found a citation to a 1967 paper of Dixon which shows that a solvable subgroup of A_n has size at most k^n where k is around 2.88.)--><br />
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8. Give an example of an abelian group A with a subgroup B such that B is not a direct summand of A.<br />
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9. Let A be the subgroup of Z^2 generated by the vector (1,0). Let B be the subgroup of Z^2 generated by the vector (3,3). Show that A and B are isomorphic to each other, but Z^2 / A is not isomorphic to Z^2 / B.<br />
<br />
==HOMEWORK 6 (due Oct 30)==<br />
<br />
1. In this problem, you will construct a functor F from the category of finite sets to the category of complex vector spaces. I'll tell you what it does on objects: if X is a finite set, then F(X) is the vector space spanned by a set of basis elements {e_x}_{x in X}, so that dim F(X) = |X|. Your job: complete the definition of F by describing the map from Mor_{Finite Sets}(X,Y) to Mor_{Vector Spaces}(F(X),F(Y)), and showing that your map respects composition.<br />
<br />
2. If (V, rho) is a representation of a group G, denote by V^G the space of vectors in V such that gv = v for all g in G; these are called _invariant_ vectors. We saw that the permutation representation of S_3 has a one-dimensional space of invariants. Prove that if X is a finite set with G-action, and V_X the corresponding permutation representation of G, then<br />
<br />
dim V_X^G = number of orbits of X.<br />
<br />
3. (Invariants are functorial) Let G be a group. Then there is a category called C[G]-Mod whose objects are complex representations of G (i.e. pairs (V,rho) where V is a complex vector space and rho is a homomorphism from G to GL(V)). The morphisms in this category are the ones described in class: Mor((V,rho),(W,psi)) is the set of linear maps f from V to W such that<br />
<br />
f(rho(g)(v)) = psi(g)(f(v))<br />
<br />
for all g in G and all v in V.<br />
<br />
Show that there is a functor H_0 from the category of representations of G to the category of vector spaces whose action on objects is given by<br />
<br />
H_0((V) = V^G.<br />
<br />
(In other words: your job again is to explain how to associate to a morphism from (V,rho) to (W,phi) a morphism from V^G to W^G, in a way that's compatible with composition.)<br />
<br />
<!--4. The group PSL_2(Z) has a presentation of the form [U,V|U^2 = V^3 = 1]. Thus, an n-dimensional representation of PSL_2(Z) is precisely a pair of n x n matrices (rho(U),rho(V)), such that rho(U)^2 = rho(V)^3 = 1. Prove that for every n > 1 there are uncountably many isomorphism classes of n-dimensional representations of PSL_2(Z). (Extra credit: show that there are uncountably many isomorphism classes of _irreducible_ representations.)<br />
<br />
An amazing theorem of Margulis (superrigidity) shows that SL_2(Z) is very special among arithmetic groups in this way. For instance, there are only _finitely_ many isomorphism classes of n-dimensional representations of PSL_d(Z) for any d > 2; those groups are "rigid."--><br />
<br />
5. Let G be a finite group, let X be a set with G-action, and let chi be the character of the corresponding permutation representation. Show that, for each g in G, chi(g) is the number of elements of X fixed by g. (I am actually going to prove this in class.)<br />
<br />
6. Let V be the space of homogeneous degree-3 polynomials in variables x_1, x_2, x_3; then S_3 acts on V by permutation of the three variables. Describe the decomposition of V into irreducible representations of S_3.<br />
<br />
7a. Show that if chi is the character of a representation of a finite group G, then chi(g^{-1}) is the complex conjugate of chi.<br />
<br />
7b. As a consequence, show that if g is conjugate to g^{-1} for all g in G, then chi(g) is real.<br />
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7c. The groups D_p and S_n have the property that g is conjugate to g^{-1} for _every_ g in G, so that every character is a real-valued function on G. Give an example of a group G where this is not the case, and give an example of a representation of G whose character takes non-real values.<br />
<br />
==HOMEWORK 7 (due Nov 6)==<br />
<br />
1. Let G be a finite group, let X be a finite set with G-action, and let chi be the character of the corresponding permutation representation. Show that, for each g in G, chi(g) is the number of elements of X fixed by g. (I am actually going to prove this in class.)<br />
<br />
2a. Show that if chi is the character of a finite-dimensional representation of a finite group G, then chi(g^{-1}) is the complex conjugate of chi(g) for all g.<br />
<br />
2b. As a consequence, show that if g is conjugate to g^{-1} for all g in G, then chi(g) is real.<br />
<br />
2c. The groups D_p and S_n have the property that g is conjugate to g^{-1} for _every_ g in G, so that every character is a real-valued function on G. Give an example of a group G where this is not the case, and give an example of a representation of G whose character takes non-real values.<br />
<br />
3. Let V and W be complex vector spaces. Let f be a linear transformation of V, and g a linear transformation of W. <br />
<br />
3a. Show that there is a unique linear transformation F satisfying<br />
<br />
F(v tensor w) = f(v) tensor g(w)<br />
<br />
for all v in V and all w in W. We denote this transformation by f tensor g.<br />
<br />
3b. Suppose V and W are finite dimensional. Show that Trace(f tensor g) = Trace(f) Trace(g). What does this say when f and g are both the identity transformation?<br />
<br />
4. Let V be a vector space. We define Sym^2 V to be the quotient of V tensor V by the subspace generated by all elements of the form<br />
<br />
v tensor w - w tensor v<br />
<br />
for v,w in V.<br />
<br />
Suppose dim V = n. What is dim Sym^2 V?<br />
<br />
5. Show that Z/pZ tensor Z/qZ is zero when p and q are distinct primes.<br />
<br />
==HOMEWORK 8 (due Nov 13)==<br />
<br />
1. (Schur's lemma can be false over non-algebraically closed fields) Let G be Z/3Z and let g be a generator of G. Then G acts by cyclic permutations on the set {1,2,3}; let P be the corresponding permutation representation on the REAL (not complex!) vector space generated by e_1,e_2,e_3.<br />
<br />
1a. Show that P breaks up as the direct sum of the trivial representation and a 2-dimensional representation V which is irreducible. (Remember, we are not working over the complex numbers, so you cannot use the criterion for irreducibility in terms of the character that we worked out in class for the complex case; you need to prove directly that V has no nontrivial proper subrepresentation.)<br />
<br />
1b. Show that the map from V to V induced by g is G-equivariant, but is neither zero nor a scalar multiplication.<br />
<br />
2. If V_1 and V_2 are representations (not necessarily irreducible!) of G, show that <chi_V_1, chi V_2> is a non-negative integer.<br />
<br />
3. When a formula always yields a non-negative integer, it is often because it is secretly a formula for either the cardinality of some finite set or the dimension of some finite-dimensional vector space. In this case, show that<br />
<br />
<chi_{V_1}, chi_{V_2}> = dim_C Hom_G(V_1,V_2)<br />
<br />
where we recall that Hom_G(V_1,G_2) refers to the space of G-equivariant homomorphisms from V_1 to V_2.<br />
<br />
4. If A is an abelian group, show that every irreducible representation of A is 1-dimensional. (Hint: show that there is a vector which is an eigenvector for every element of A!)<br />
<br />
5. Let G be the group of order 20 which is the semidirect product of A by H, where A is Z/5Z and H is (Z/5Z)^* with its natural action on A. We will work out all the irreducible representations of A. <br />
<br />
5a. First of all, consider the 5-dimensional permutation representation of G acting by left multiplication on the cosets of H. Show that this representation decomposes as a direct sum of the trivial representation and an irreducible 4-dimensional representation. (Hint: it is probably easier to use the character of this representation than to prove irreducibility directly.) <br />
<br />
5b. Now show that there are 4 1-dimensional representations of G which factor through the quotient G/A (i.e. such that rho(A) is the identity.)<br />
<br />
5c. Using the decomposition of the regular representation, show that the irreducible representations of G you have constructed are the only ones.<br />
<br />
==HOMEWORK 9 (due Nov 20)==<br />
<br />
1. If R is a ring (not necessarily with 1) such that a^2 = a for every a in R, R is called a Boolean ring.<br />
<br />
1a. Show that every Boolean ring is commutative.<br />
<br />
1b. Show that any nonzero Boolean ring has characteristic 2. (In other words, show that a+a=0 for all a in R.)<br />
<br />
<br />
2. Let phi : R -> S be a homomorphism of commutative rings with 1.<br />
<br />
2a. If P is a prime ideal of S, show that phi^(-1) (P) is either R or a prime ideal of R.<br />
<br />
2b. If M is a maximal ideal of S and phi is surjective, show that phi^(-1) (M) is a maximal ideal of R.<br />
<br />
2c. Find a homomorphism phi : R -> S and a maximal ideal M of S such that phi^(-1) (M) is not a maximal ideal of R.<br />
<br />
<br />
3. Let Nil(A) denote the set of nilpotent elements of a ring A, and let R be a commutative ring. In class it was observed that Nil(R) is an ideal of R.<br />
<br />
3a. Find (a generator of) Nil(Z/720Z).<br />
<br />
3b. Show that Nil(R/Nil(R)) = 0.<br />
<br />
3c. If x is in Nil(R) and u is a unit, show that u+x is a unit.<br />
<br />
3d. Show that Nil(S), where S is the 2x2 matrices with real coefficients, is not an ideal of S.<br />
<br />
<br />
4a. If M is an R-module and I is an ideal of R such that im = 0 for all i in I and m in M, show that M carries the structure of an R/I-module.<br />
<br />
4b. If S is a commutative ring with 1 and J is a maximal ideal of S, show that J/J^2 is a vector space over S/J.<br />
<br />
<br />
5. A "divisible element" x of a Z-module is one such that for each positive integer k, there exists a y such that ky=x. Show that a free Z-module cannot contain a nonzero divisible element.<br />
<br />
<br />
6. Prove that the direct sum of any collection of free R-modules is also a free R-module.<br />
<br />
<br />
7. In this problem you will prove that the module M given by the direct product of copies of Z, indexed by the positive integers, is not a free Z-module. Therefore, assume that M is a free Z-module with basis B, and let N be the direct sum of copies of Z, as a submodule of M.<br />
<br />
7a. Show there is a countable subset B1 of B such that N is contained in the submodule N1 generated by B1. [Hint: N is countable.]<br />
<br />
7b. Show that M/N1 is a free Z-module.<br />
<br />
7c. Let S = {(b1, b2, b3, ...) : b_i = i! or -i!}. Show that there is some s in S which is not in N1. [Hint: S is uncountable.]<br />
<br />
7d. For any s in S but not N1, show that for each positive integer k, there exists some m in M with s-km in N1. Conclude that s+N1 is a divisible element of M/N1.<br />
<br />
7e. Use problem 5 to obtain a contradiction.<br />
<br />
<br />
8. In class I said "rank" is not always well-defined for R-modules if R is not commutative. Here is an example demonstrating this assertion. Let M be the countably infinite direct product of copies of Z as in exercise 7, and let R be the endomorphism ring of M. Define phi_1(a1,a2,a3,a4,...) = (a1,a3,a5,...) and phi_2(a1,a2,a3,a4,...) = (a2,a4,a6,...).<br />
<br />
8a. Show that {phi_1, phi_2} is a free basis for R, as a left R-module. [Hint: let psi_1(a1,a2,a3,...) = (a1,0,a2,0,...) and psi_2(a1,a2,a3,...) = (0,a1,0,a2,...). Compute the products of the phi_i and psi_j in either order, and use the result to show that the phi_i are a basis.]<br />
<br />
8b. Show that R is isomorphic to R^n for every positive integer n.</div>Dummithttps://hilbert.math.wisc.edu/wiki/index.php?title=741&diff=47137412012-11-15T20:11:57Z<p>Dummit: /* HOMEWORK 9 (due Nov 20) */</p>
<hr />
<div>'''Math 741'''<br />
<br />
Algebra<br />
<br />
Prof: [http://www.math.wisc.edu/~ellenber Jordan Ellenberg]<br />
<br />
Grader: Evan Dummit<br />
<br />
Ellenberg's office hours: Friday 3pm.<br />
<br />
Grader's office hours: Monday 4pm. Late homework may be given directly to the grader, along with either (i) the instructor's permission, or (ii) a polite request for mercy.<br />
<br />
This course, the first semester of the introductory graduate sequence in algebra, will cover the basic theory of groups, group actions, representation, linear and multilinear algebra, and the beginnings of ring theory.<br />
<br />
==SYLLABUS==<br />
<br />
In this space we will record the theorems and definitions we covered each week, which we can use as a list of notions you should be prepared to answer questions about on the Algebra qualifying exam. The material covered on the homework is also an excellent guide to the scope of the course. <br />
<br />
'''WEEK 1''': <br />
<br />
Definition of group. Associativity. Inverse. <br />
<br />
Examples of group: GL_n(R). GL_n(Z). Z/nZ. R. Z. R^*. The free group F_k on k generators. <br />
<br />
Homomorphisms. The homomorphisms from F_k to G are in bijection with G^k. Isomorphisms.<br />
<br />
'''WEEK 2''':<br />
<br />
The symmetric group (or permutation group) S_n on n letters. Cycle decomposition of a permutation. Order of a permutation. Thm: every element of a finite group has finite order. <br />
<br />
Subgroups. Left and right cosets. Lagrange's Theorem. Cyclic groups. The order of an element of a finite group is a divisor of the order of the group.<br />
<br />
The sign homomorphism S_n -> +-1.<br />
<br />
<br />
'''WEEK 3'''<br />
<br />
Normal subgroups. The quotient of a group by a normal subgroup. The first isomorphism theorem. Examples of S_n -> +-1 and S_4 -> S_3 with kernel V_4, the Klein 4-group.<br />
<br />
Centralizers and centers. Abelian groups. The center of SL_n(R) is either 1 or +-1.<br />
<br />
Groups with presentations. The infinite dihedral group <x,y | x^2 = 1, y^2 = 1>.<br />
<br />
'''WEEK 4'''<br />
<br />
More on groups with presentations.<br />
<br />
Second and third isomorphism theorems.<br />
<br />
Semidirect products.<br />
<br />
'''WEEK 5'''<br />
<br />
Group actions, orbits, and stabilizers.<br />
<br />
Orbit-stabilizer theorem.<br />
<br />
Cayley's theorem.<br />
<br />
Cauchy's theorem.<br />
<br />
'''WEEK 6'''<br />
<br />
Applications of orbit-stabilizer theorem (p-groups have nontrivial center, first Sylow theorem.)<br />
<br />
Classification of finite abelian groups and finitely generated abelian groups.<br />
<br />
Composition series and the Jordan-Holder theorem (which we state but don't prove.) <br />
<br />
The difference between knowing the composition factors and knowing the group (e.g. all p-groups of the same order have the same composition factors.)<br />
<br />
'''WEEK 7'''<br />
<br />
Simplicity of A_n.<br />
<br />
Nilpotent groups (main example: the Heisenberg group)<br />
<br />
Derived series and lower central series.<br />
<br />
Category theory 101: Definition of category and functor. Some examples. A group is a groupoid with one object.<br />
<br />
'''WEEK 8'''<br />
<br />
Introduction to representation theory. <br />
<br />
<br />
<br />
Below you will find a repository of homework problems. Note that some of these problems are taken from Lang's ''Algebra''.<br />
<br />
==HOMEWORK 1 (due Sep 20)==<br />
<br />
1. Suppose that H_1 and H_2 are subgroups of a group G. Prove that the intersection of H_1 and H_2 is a subgroup of G.<br />
<br />
2. Recall that S_3 (the symmetric group) is the group of permutations of the set {1..3}. List all the subgroups of S_3.<br />
<br />
3. We can define an equivalence relation on rational numbers by declaring two rational numbers to be equal whenever they differ by an integer. We denote the set of equivalence classes by Q/Z. The operation of addition makes Q/Z into a group.<br />
<br />
a) For each n, prove that Q/Z has a subgroup of order n.<br />
<br />
b) Prove that Q/Z is a ''divisible'' group: that is, if x is an element of Q/Z and n is an integer, there exists an element y of Q/Z such that ny = x. (Note that we write the operation in this group as addition rather than multiplication, which is why we write ny for the n-fold product of y with itself rather than y^n)<br />
<br />
c) Prove that Q/Z is not finitely generated. (Hint: prove that if x_1, .. x_d is a finite subset of Q/Z, the subgroup of Q/Z generated by x_1, ... x_d is finite.)<br />
<br />
4. Let F_2 be the free group on two generators x,y. Prove that there is a automorphism of F_2 which sends x to xyx and y to xy, and prove that this automorphism is unique.<br />
<br />
5. Let H be a subgroup of G, and let N_G(H), the ''normalizer'' of H in G, be the set of elements g in G satsifying g H g^{-1} = H. Prove that N_G(H) is a subgroup of G.<br />
<br />
6. Let T be the subgroup of GL_2(R) consisting of diagonal matrices. This is an example of a "Cartan subgroup," or "torus" (whence the notation T.) Describe the normalizer N(T) of T in GL_2(R). Prove that there is a homomorphism from N(T) to Z/2Z whose kernel is precisely T. <br />
<br />
7. Prove that the group of inner automorphisms of a group is normal in the full automorphism group Aut(G).<br />
<br />
8. Let H and H' be subgroups of a group G, and let x be an element of G. We denote by HxH' the set of elements of the form hxh', with h in H and h' in H', and we call HxH a ''double coset'' of the pair (H,H').<br />
<br />
a) Show that G decomposes as a disjoint union of double cosets of (H,H').<br />
<br />
b) Let G be GL_2(R) and let B be the subgroup of upper triangular matrices. Prove that G decomposes into exactly two double cosets of (B,B).<br />
<br />
c) Let G be the symmetric group S_n, and let H be the subgroup of permutations which fix 1. Describe the double coset decomposition of S_n into double cosets of (H,H).<br />
<br />
==HOMEWORK 2 (due Sep 27)==<br />
<br />
1. Prove that the symmetric group S_n is generated by transpositions. Hint: proceed by induction on n. Let H be the subgroup of S_n consisting of permutations fixing 1. Observe that H is isomorphic to S_{n-1}, so that H is generated by transpositions. Explain why it suffices to show that, for any g in S_n, the coset gH contains a transposition. Then explain why this is the case.<br />
<br />
2. Given that the symmetric group S_n is generated by transpositions, show that there are no nontrivial homomorphisms from S_n to Z/pZ for any odd prime p.<br />
<br />
3. Compute the centralizer of the triple flip (12)(34)(56) in S_6. (Note: this is the kind of problem that a computer algebra package like sage is very good at, and I encourage you to learn to use sage well enough to check your answer.)<br />
<br />
4. Warning: a normal subgroup of a normal subgroup need not be normal! Let H be the subgroup of S_4 generated by (12)(34). Then H is a subgroup of the Klein 4-group V_4, which we have already shown is normal in S_4. Show that H is normal in V_4, but H is not normal in S_4.<br />
<br />
5. The '''quaternion group''' Q is a non-abelian finite group of order 8. Its elements are 1, -1, i, -i, j, -j, k, -k; it satisfies g^2 = -1 for all g except +-1, and ijk = 1. (Note that the notation "-i" means "the product of i with -1" and -1 is understood to be central in Q.) Prove that every subgroup of Q is normal.<br />
<br />
6. (finishing example done in class) Let Gamma be the group F<x,y> / (x^2 = 1, y^2 = 1). (Remember, this means the quotient of the free group by the group generated by all conjugates of x^2 and y^2.) Prove that Gamma is infinite. Prove that Gamma has a subgroup R generated by xy isomorphic to Z, that R is normal, and that the quotient Gamma/R is isomorphic to Z/2Z. <br />
<br />
7. Suppose a group G has the property that every element in g satisfies g^2 = 1. Show that G must be abelian.<br />
<br />
==HOMEWORK 3 (due Oct 4)==<br />
<br />
1. Multiplication by -1 is an automorphism of Z/nZ. Let a be the homomorphism Z/2Z -> Aut(Z/nZ) which sends the nontrivial element of Z/2Z to multiplication by -1. Then we have a semidirect product of Z/pZ by Z/2Z as defined in class. This group of order 2n is called a ''dihedral group.'' and is denoted D_n (or sometimes D_{2n}).<br />
<br />
1a. Compute the center of D_n. (Note that the answer depends on n!)<br />
1b. Show that all involutions of D_n are conjugate to each other if and only if n is odd.<br />
<br />
2. (More on direct products that we didn't do in class.) Suppose that G is a semidirect product of N with H. Suppose furthermore that H is also normal in G. Prove that G is a direct product of N with H.<br />
<br />
3. Let Q be the quaternion group from last week's homework. Show that Q does not decompose as a semidirect product N \rtimes H, apart form the trivial cases N = {e} and N = Q.<br />
<br />
4. The ''affine linear group'' of degree n is the group of transformations from R^n to R^n of the form x -> Ax + b, where A is a matrix in GL_n(R) and b is a vector in R^n. Prove that the affine linear group is a semidirect product R^n \rtimes GL_n.<br />
<br />
5. The ''ordinary triangle group'' T(p,q,r) is the group with presentation <x,y | x^p = y^q = (xy)^r = 1>. This is a very interesting family of groups which arises naturally in many contexts in number theory and topology.<br />
<br />
6a. Prove that T(2,2,n) is isomorphic to the dihedral group D_n.<br />
6b. Prove that T(2,3,3) is finite and compute its order. Is it isomorphic to a finite group we've discussed in class?<br />
<br />
(It is a beautiful fact, but outside the scope of this course, that T(p,q,r) is finite if and only if 1/p + 1/q + 1/r > 1.)<br />
<br />
7. Let G be a subgroup of GL_n(R) which contains SL_n(R). Prove that G is a normal subgroup of GL_n(R).<br />
<br />
8. Let G be a group and let H_1 and H_2 be subgroups of G of finite index (that is, the number of cosets in G/H_i is finite, i = 1,2.) Prove that H_1 intersect H_2 also has finite index in G.<br />
<br />
9. Recall that the commutator [x,y] is x y x^{-1} y^{-1}. If G is a group, the commutator subgroup of G, denoted G' or [G,G], is the subgroup of G generated by all commutators [x,y] for x,y in G.<br />
<br />
9a. Show that G' is a normal subgroup of G.<br />
9b. Show that G/G' is an abelian group.<br />
9c. Show that if f: G -> A is a homomorphism, with A abelian, then G' is contained in ker f. Conclude that f factors through a homomorphism G/G' -> A.<br />
9d. Show that G' = G if and only if G has no nontrivial abelian quotient. In this case, we say G is ''perfect''.<br />
<br />
10. Under what conditions on (p,q,r) is the triangle group T(p,q,r) perfect?<br />
<br />
==HOMEWORK 4 (due Oct 16)==<br />
<br />
1. Let X be a set with a transitive action of a group G, which we think of as a homomorphism f: G -> Sym(X). Let H_x be the stabilizer of an element x of X.<br />
<br />
1a. If x' is another element of X, show that H_{x"} and H_x are conjugate subgroups. (Hint: this is not true without the hypothesis of transitivity, so make sure to use it!)<br />
<br />
1b. Show that the kernel of f is the intersection of all the conjugates of H_x in G. (This subgroup is called the ''normal core'' of H_x.)<br />
<br />
2. This week, a certified mathematical grown-up asked on MathOverflow what the index-3 subgroups of SL_2(Z/pZ) were. Let's do it ourselves for homework. Let p be a prime, and suppose H is an index-3 subgroup of SL_2(Z/pZ). I do not claim that the proof sketched below is necessarily the easiest, but it allows us to cover some ground that we won't get to do in lecture.<br />
<br />
2a. Using the coset action, show that the normal core of H is the kernel of a homomorphism f from SL_2(Z/pZ) to S_3.<br />
<br />
2b. A ''unipotent matrix'' is a matrix A such that (A-I)^n = 0 for some positive integer n. Prove that a unipotent element of SL_2(Z/pZ) has order dividing p.<br />
<br />
2c. You may use the fact that unipotent matrices generate SL_2(Z/pZ). Given this fact, show that f must be trivial when p > 3.<br />
<br />
2d. Using the above, show that SL_2(Z/pZ) has no index-3 subgroups when p > 3.<br />
<br />
2e. (extra credit) What can you say about the minimal index of a proper subgroup of SL_2(Z/pZ)?<br />
<br />
3. Let H be a subgroup of G of index 2. Prove that H is normal.<br />
<br />
4. We proved two theorems in class: Lagrange's theorem, which says that the order of a subgroup of G divides |G|, and Cauchy's theorem, which says that if p divides |G| then there exists a subgroup of G of order p. Cauchy's theorem is a kind of partial converse to Lagrange's theorem, but the full converse is false. Give an example of a finite group G and a divisor n of |G| such that you can prove there is no subgroup of G of order n.<br />
<br />
5. Let X be the set of lines in F_3^2 (here a line means a 1-dimensional linear subspace) so that |X| = 4. Let G be the projective linear group PGL_2(F_3), so that G acts on X via its action on F_3^2. This action can be thought of as a homomorphism from G to S_4. Show that this homomorphism is an ISOMORPHISM from PGL_2(F_3) to S_4.<br />
<br />
6. The group S_3 has only three conjugacy classes. Prove that a finite group with at most three conjugacy classes has order at most 6.<br />
<br />
7. Let X be the set of ordered triples of elements of {1,..,n}, for some n >= 3. Then S_n acts on X. How many orbits are there? (Hint: the answer does not depend on n.)<br />
<br />
8. Suppose that G and H are groups and f: G -> H is a homomorphism. Recall the definition of the abelianization G^ab from the previous problem set.<br />
<br />
8a. Show that the composition G -> H -> H^ab factors through a unique homomorphism G^ab -> H^ab, which we denote f^ab.<br />
<br />
8b. Show that if f: G -> H and g: H -> Q are homomorphisms, then f^ab o g^ab = (f o g)^ab.<br />
<br />
(For those reading MacLane, this constitutes a proof that abelianization is a '''functor''' from the category of groups to the category of abelian groups.)<br />
<br />
==HOMEWORK 5 (due Oct 23)==<br />
<br />
1. Show that there are only two different isomorphism classes of groups of order 14. (Hint: prove that such a group must be a semidirect product of a group of order 7 by a group of order 2.)<br />
<br />
2. (variant of Burnside's Lemma) Let G be a finite group and H a normal subgroup. Suppose G acts on a finite set X. <br />
<br />
2a. Show that if x and y lie in the same H-orbit, and g is an element of G, then gx and gy also lie in the same H-orbit. Thus, G/H has a natural action on the set of H-orbits X/H.<br />
<br />
2b. Show that the average number of fixed points in X of an element of the coset gH is the same as the number of fixed points of the element gH of G/H in its action on X/H.<br />
<br />
3. A ''central extension'' of a group G by an abelian group A is a group E, together with a surjective homomorphism E -> G whose kernel is central in E and is isomorphic to A.<br />
<br />
3a. Show that the quaternion group is a central extension of (Z/2Z)^2 by Z/2Z.<br />
<br />
3b. Give an example of another nonabelian group which is a central extension of (Z/2Z)^2 by Z/2Z and which is not isomorphic to the quaternion group.<br />
<br />
4. Prove that a central extension of an abelian group is nilpotent.<br />
<br />
5. Give two different composition series for S_4 and show that they have the same composition factors.<br />
<br />
6. Suppose that A is a finitely generated abelian group, and B is a subgroup of A.<br />
Prove that rank(A/B) = rank(A) - rank(B).<br />
ADDED: I asserted in class, but did not prove, that a subgroup of a finitely generated abelian group is again finitely generated. As one class member pointed out, without this fact it's not even clear why it makes sense to talk about rank(B)! You may assume it (though it would have been better for me to have included proving this fact as part of the problem....)<br />
<br />
7. For this problem, you may use the fact (which I should have asked on homework earlier) that if H_1 and H_2 are subgroups of G, then<br />
<br />
[G: H_1 intersect H_2] <= [G:H_1][G:H_2].<br />
<br />
Prove that if G is a simple group, d is an integer greater than 1, and S is a subgroup of index d in G, then |G| <= d^d.<br />
<br />
Conclude that, in particular, A_5 has no subgroup of index 3. On the other hand, exhibit a subgroup of A_5 of index 5.<br />
<br />
(Note: d^d is not a sharp bound; can you get a bound of d factorial instead?)<br />
<br />
<!--(This argument shows that a solvable subgroup of A_n has size at most about (n-1)!, but this is far from sharp; e.g. I found a citation to a 1967 paper of Dixon which shows that a solvable subgroup of A_n has size at most k^n where k is around 2.88.)--><br />
<br />
8. Give an example of an abelian group A with a subgroup B such that B is not a direct summand of A.<br />
<br />
9. Let A be the subgroup of Z^2 generated by the vector (1,0). Let B be the subgroup of Z^2 generated by the vector (3,3). Show that A and B are isomorphic to each other, but Z^2 / A is not isomorphic to Z^2 / B.<br />
<br />
==HOMEWORK 6 (due Oct 30)==<br />
<br />
1. In this problem, you will construct a functor F from the category of finite sets to the category of complex vector spaces. I'll tell you what it does on objects: if X is a finite set, then F(X) is the vector space spanned by a set of basis elements {e_x}_{x in X}, so that dim F(X) = |X|. Your job: complete the definition of F by describing the map from Mor_{Finite Sets}(X,Y) to Mor_{Vector Spaces}(F(X),F(Y)), and showing that your map respects composition.<br />
<br />
2. If (V, rho) is a representation of a group G, denote by V^G the space of vectors in V such that gv = v for all g in G; these are called _invariant_ vectors. We saw that the permutation representation of S_3 has a one-dimensional space of invariants. Prove that if X is a finite set with G-action, and V_X the corresponding permutation representation of G, then<br />
<br />
dim V_X^G = number of orbits of X.<br />
<br />
3. (Invariants are functorial) Let G be a group. Then there is a category called C[G]-Mod whose objects are complex representations of G (i.e. pairs (V,rho) where V is a complex vector space and rho is a homomorphism from G to GL(V)). The morphisms in this category are the ones described in class: Mor((V,rho),(W,psi)) is the set of linear maps f from V to W such that<br />
<br />
f(rho(g)(v)) = psi(g)(f(v))<br />
<br />
for all g in G and all v in V.<br />
<br />
Show that there is a functor H_0 from the category of representations of G to the category of vector spaces whose action on objects is given by<br />
<br />
H_0((V) = V^G.<br />
<br />
(In other words: your job again is to explain how to associate to a morphism from (V,rho) to (W,phi) a morphism from V^G to W^G, in a way that's compatible with composition.)<br />
<br />
<!--4. The group PSL_2(Z) has a presentation of the form [U,V|U^2 = V^3 = 1]. Thus, an n-dimensional representation of PSL_2(Z) is precisely a pair of n x n matrices (rho(U),rho(V)), such that rho(U)^2 = rho(V)^3 = 1. Prove that for every n > 1 there are uncountably many isomorphism classes of n-dimensional representations of PSL_2(Z). (Extra credit: show that there are uncountably many isomorphism classes of _irreducible_ representations.)<br />
<br />
An amazing theorem of Margulis (superrigidity) shows that SL_2(Z) is very special among arithmetic groups in this way. For instance, there are only _finitely_ many isomorphism classes of n-dimensional representations of PSL_d(Z) for any d > 2; those groups are "rigid."--><br />
<br />
5. Let G be a finite group, let X be a set with G-action, and let chi be the character of the corresponding permutation representation. Show that, for each g in G, chi(g) is the number of elements of X fixed by g. (I am actually going to prove this in class.)<br />
<br />
6. Let V be the space of homogeneous degree-3 polynomials in variables x_1, x_2, x_3; then S_3 acts on V by permutation of the three variables. Describe the decomposition of V into irreducible representations of S_3.<br />
<br />
7a. Show that if chi is the character of a representation of a finite group G, then chi(g^{-1}) is the complex conjugate of chi.<br />
<br />
7b. As a consequence, show that if g is conjugate to g^{-1} for all g in G, then chi(g) is real.<br />
<br />
7c. The groups D_p and S_n have the property that g is conjugate to g^{-1} for _every_ g in G, so that every character is a real-valued function on G. Give an example of a group G where this is not the case, and give an example of a representation of G whose character takes non-real values.<br />
<br />
==HOMEWORK 7 (due Nov 6)==<br />
<br />
1. Let G be a finite group, let X be a finite set with G-action, and let chi be the character of the corresponding permutation representation. Show that, for each g in G, chi(g) is the number of elements of X fixed by g. (I am actually going to prove this in class.)<br />
<br />
2a. Show that if chi is the character of a finite-dimensional representation of a finite group G, then chi(g^{-1}) is the complex conjugate of chi(g) for all g.<br />
<br />
2b. As a consequence, show that if g is conjugate to g^{-1} for all g in G, then chi(g) is real.<br />
<br />
2c. The groups D_p and S_n have the property that g is conjugate to g^{-1} for _every_ g in G, so that every character is a real-valued function on G. Give an example of a group G where this is not the case, and give an example of a representation of G whose character takes non-real values.<br />
<br />
3. Let V and W be complex vector spaces. Let f be a linear transformation of V, and g a linear transformation of W. <br />
<br />
3a. Show that there is a unique linear transformation F satisfying<br />
<br />
F(v tensor w) = f(v) tensor g(w)<br />
<br />
for all v in V and all w in W. We denote this transformation by f tensor g.<br />
<br />
3b. Suppose V and W are finite dimensional. Show that Trace(f tensor g) = Trace(f) Trace(g). What does this say when f and g are both the identity transformation?<br />
<br />
4. Let V be a vector space. We define Sym^2 V to be the quotient of V tensor V by the subspace generated by all elements of the form<br />
<br />
v tensor w - w tensor v<br />
<br />
for v,w in V.<br />
<br />
Suppose dim V = n. What is dim Sym^2 V?<br />
<br />
5. Show that Z/pZ tensor Z/qZ is zero when p and q are distinct primes.<br />
<br />
==HOMEWORK 8 (due Nov 13)==<br />
<br />
1. (Schur's lemma can be false over non-algebraically closed fields) Let G be Z/3Z and let g be a generator of G. Then G acts by cyclic permutations on the set {1,2,3}; let P be the corresponding permutation representation on the REAL (not complex!) vector space generated by e_1,e_2,e_3.<br />
<br />
1a. Show that P breaks up as the direct sum of the trivial representation and a 2-dimensional representation V which is irreducible. (Remember, we are not working over the complex numbers, so you cannot use the criterion for irreducibility in terms of the character that we worked out in class for the complex case; you need to prove directly that V has no nontrivial proper subrepresentation.)<br />
<br />
1b. Show that the map from V to V induced by g is G-equivariant, but is neither zero nor a scalar multiplication.<br />
<br />
2. If V_1 and V_2 are representations (not necessarily irreducible!) of G, show that <chi_V_1, chi V_2> is a non-negative integer.<br />
<br />
3. When a formula always yields a non-negative integer, it is often because it is secretly a formula for either the cardinality of some finite set or the dimension of some finite-dimensional vector space. In this case, show that<br />
<br />
<chi_{V_1}, chi_{V_2}> = dim_C Hom_G(V_1,V_2)<br />
<br />
where we recall that Hom_G(V_1,G_2) refers to the space of G-equivariant homomorphisms from V_1 to V_2.<br />
<br />
4. If A is an abelian group, show that every irreducible representation of A is 1-dimensional. (Hint: show that there is a vector which is an eigenvector for every element of A!)<br />
<br />
5. Let G be the group of order 20 which is the semidirect product of A by H, where A is Z/5Z and H is (Z/5Z)^* with its natural action on A. We will work out all the irreducible representations of A. <br />
<br />
5a. First of all, consider the 5-dimensional permutation representation of G acting by left multiplication on the cosets of H. Show that this representation decomposes as a direct sum of the trivial representation and an irreducible 4-dimensional representation. (Hint: it is probably easier to use the character of this representation than to prove irreducibility directly.) <br />
<br />
5b. Now show that there are 4 1-dimensional representations of G which factor through the quotient G/A (i.e. such that rho(A) is the identity.)<br />
<br />
5c. Using the decomposition of the regular representation, show that the irreducible representations of G you have constructed are the only ones.<br />
<br />
==HOMEWORK 9 (due Nov 20)==<br />
<br />
1. If R is a ring (not necessarily with 1) such that a^2 = a for every a in R, R is called a Boolean ring.<br />
<br />
1a. Show that every Boolean ring is commutative.<br />
<br />
1b. Show that any nonzero Boolean ring has characteristic 2. (In other words, show that a+a=0 for all a in R.)<br />
<br />
<br />
2. Let phi : R -> S be a homomorphism of commutative rings with 1.<br />
<br />
2a. If P is a prime ideal of S, show that phi^(-1) (P) is either R or a prime ideal of R.<br />
<br />
2b. If M is a maximal ideal of S and phi is surjective, show that phi^(-1) (M) is a maximal ideal of R.<br />
<br />
2c. Find a homomorphism phi : R -> S and a maximal ideal M of S such that phi^(-1) (M) is not a maximal ideal of R.<br />
<br />
<br />
3. Let Nil(A) denote the set of nilpotent elements of a ring A, and let R be a commutative ring. In class it was observed that Nil(R) is an ideal of R.<br />
<br />
3a. Find (a generator of) Nil(Z/720Z).<br />
<br />
3b. Show that Nil(R/Nil(R)) = 0.<br />
<br />
3c. If x is in Nil(R) and u is a unit, show that u+x is a unit.<br />
<br />
3d. Show that Nil(S), where S is the 2x2 matrices with real coefficients, is not an ideal of S.<br />
<br />
<br />
4a. If M is an R-module and I is an ideal of R such that im = 0 for all i in I and m in M, show that M carries the structure of an R/I-module.<br />
<br />
4b. If S is a commutative ring with 1 and J is a maximal ideal of S, show that J/J^2 is a vector space over S/J.<br />
<br />
<br />
5. A "divisible element" x of a Z-module is one such that for each positive integer k, there exists a y such that ky=x. Show that a free Z-module cannot contain a divisible element.<br />
<br />
<br />
6. Prove that the direct sum of any collection of free R-modules is also a free R-module.<br />
<br />
<br />
7. In this problem you will prove that the module M given by the direct product of copies of Z, indexed by the positive integers, is not a free Z-module. Therefore, assume that M is a free Z-module with basis B, and let N be the direct sum of copies of Z, as a submodule of M.<br />
<br />
7a. Show there is a countable subset B1 of B such that N is contained in the submodule N1 generated by B1. [Hint: N is countable.]<br />
<br />
7b. Show that M/N1 is a free Z-module.<br />
<br />
7c. Let S = {(b1, b2, b3, ...) : b_i = i! or -i!}. Show that there is some s in S which is not in N1. [Hint: S is uncountable.]<br />
<br />
7d. For any s in S but not N1, show that for each positive integer k, there exists some m in M with s-km in N1. Conclude that s+N1 is a divisible element of M/N1.<br />
<br />
7e. Use problem 5 to obtain a contradiction.<br />
<br />
<br />
8. In class I said "rank" is not always well-defined for R-modules if R is not commutative. Here is an example demonstrating this assertion. Let M be the countably infinite direct product of copies of Z as in exercise 7, and let R be the endomorphism ring of M. Define phi_1(a1,a2,a3,a4,...) = (a1,a3,a5,...) and phi_2(a1,a2,a3,a4,...) = (a2,a4,a6,...).<br />
<br />
8a. Show that {phi_1, phi_2} is a free basis for R, as a left R-module. [Hint: let psi_1(a1,a2,a3,...) = (a1,0,a2,0,...) and psi_2(a1,a2,a3,...) = (0,a1,0,a2,...). Compute the products of the phi_i and psi_j in either order, and use the result to show that the phi_i are a basis.]<br />
<br />
8b. Show that R is isomorphic to R^n for every positive integer n.</div>Dummithttps://hilbert.math.wisc.edu/wiki/index.php?title=741&diff=47127412012-11-15T05:27:05Z<p>Dummit: /* HOMEWORK 9 (due Nov 20) */ typo fix</p>
<hr />
<div>'''Math 741'''<br />
<br />
Algebra<br />
<br />
Prof: [http://www.math.wisc.edu/~ellenber Jordan Ellenberg]<br />
<br />
Grader: Evan Dummit<br />
<br />
Ellenberg's office hours: Friday 3pm.<br />
<br />
Grader's office hours: Monday 4pm. Late homework may be given directly to the grader, along with either (i) the instructor's permission, or (ii) a polite request for mercy.<br />
<br />
This course, the first semester of the introductory graduate sequence in algebra, will cover the basic theory of groups, group actions, representation, linear and multilinear algebra, and the beginnings of ring theory.<br />
<br />
==SYLLABUS==<br />
<br />
In this space we will record the theorems and definitions we covered each week, which we can use as a list of notions you should be prepared to answer questions about on the Algebra qualifying exam. The material covered on the homework is also an excellent guide to the scope of the course. <br />
<br />
'''WEEK 1''': <br />
<br />
Definition of group. Associativity. Inverse. <br />
<br />
Examples of group: GL_n(R). GL_n(Z). Z/nZ. R. Z. R^*. The free group F_k on k generators. <br />
<br />
Homomorphisms. The homomorphisms from F_k to G are in bijection with G^k. Isomorphisms.<br />
<br />
'''WEEK 2''':<br />
<br />
The symmetric group (or permutation group) S_n on n letters. Cycle decomposition of a permutation. Order of a permutation. Thm: every element of a finite group has finite order. <br />
<br />
Subgroups. Left and right cosets. Lagrange's Theorem. Cyclic groups. The order of an element of a finite group is a divisor of the order of the group.<br />
<br />
The sign homomorphism S_n -> +-1.<br />
<br />
<br />
'''WEEK 3'''<br />
<br />
Normal subgroups. The quotient of a group by a normal subgroup. The first isomorphism theorem. Examples of S_n -> +-1 and S_4 -> S_3 with kernel V_4, the Klein 4-group.<br />
<br />
Centralizers and centers. Abelian groups. The center of SL_n(R) is either 1 or +-1.<br />
<br />
Groups with presentations. The infinite dihedral group <x,y | x^2 = 1, y^2 = 1>.<br />
<br />
'''WEEK 4'''<br />
<br />
More on groups with presentations.<br />
<br />
Second and third isomorphism theorems.<br />
<br />
Semidirect products.<br />
<br />
'''WEEK 5'''<br />
<br />
Group actions, orbits, and stabilizers.<br />
<br />
Orbit-stabilizer theorem.<br />
<br />
Cayley's theorem.<br />
<br />
Cauchy's theorem.<br />
<br />
'''WEEK 6'''<br />
<br />
Applications of orbit-stabilizer theorem (p-groups have nontrivial center, first Sylow theorem.)<br />
<br />
Classification of finite abelian groups and finitely generated abelian groups.<br />
<br />
Composition series and the Jordan-Holder theorem (which we state but don't prove.) <br />
<br />
The difference between knowing the composition factors and knowing the group (e.g. all p-groups of the same order have the same composition factors.)<br />
<br />
'''WEEK 7'''<br />
<br />
Simplicity of A_n.<br />
<br />
Nilpotent groups (main example: the Heisenberg group)<br />
<br />
Derived series and lower central series.<br />
<br />
Category theory 101: Definition of category and functor. Some examples. A group is a groupoid with one object.<br />
<br />
'''WEEK 8'''<br />
<br />
Introduction to representation theory. <br />
<br />
<br />
<br />
Below you will find a repository of homework problems. Note that some of these problems are taken from Lang's ''Algebra''.<br />
<br />
==HOMEWORK 1 (due Sep 20)==<br />
<br />
1. Suppose that H_1 and H_2 are subgroups of a group G. Prove that the intersection of H_1 and H_2 is a subgroup of G.<br />
<br />
2. Recall that S_3 (the symmetric group) is the group of permutations of the set {1..3}. List all the subgroups of S_3.<br />
<br />
3. We can define an equivalence relation on rational numbers by declaring two rational numbers to be equal whenever they differ by an integer. We denote the set of equivalence classes by Q/Z. The operation of addition makes Q/Z into a group.<br />
<br />
a) For each n, prove that Q/Z has a subgroup of order n.<br />
<br />
b) Prove that Q/Z is a ''divisible'' group: that is, if x is an element of Q/Z and n is an integer, there exists an element y of Q/Z such that ny = x. (Note that we write the operation in this group as addition rather than multiplication, which is why we write ny for the n-fold product of y with itself rather than y^n)<br />
<br />
c) Prove that Q/Z is not finitely generated. (Hint: prove that if x_1, .. x_d is a finite subset of Q/Z, the subgroup of Q/Z generated by x_1, ... x_d is finite.)<br />
<br />
4. Let F_2 be the free group on two generators x,y. Prove that there is a automorphism of F_2 which sends x to xyx and y to xy, and prove that this automorphism is unique.<br />
<br />
5. Let H be a subgroup of G, and let N_G(H), the ''normalizer'' of H in G, be the set of elements g in G satsifying g H g^{-1} = H. Prove that N_G(H) is a subgroup of G.<br />
<br />
6. Let T be the subgroup of GL_2(R) consisting of diagonal matrices. This is an example of a "Cartan subgroup," or "torus" (whence the notation T.) Describe the normalizer N(T) of T in GL_2(R). Prove that there is a homomorphism from N(T) to Z/2Z whose kernel is precisely T. <br />
<br />
7. Prove that the group of inner automorphisms of a group is normal in the full automorphism group Aut(G).<br />
<br />
8. Let H and H' be subgroups of a group G, and let x be an element of G. We denote by HxH' the set of elements of the form hxh', with h in H and h' in H', and we call HxH a ''double coset'' of the pair (H,H').<br />
<br />
a) Show that G decomposes as a disjoint union of double cosets of (H,H').<br />
<br />
b) Let G be GL_2(R) and let B be the subgroup of upper triangular matrices. Prove that G decomposes into exactly two double cosets of (B,B).<br />
<br />
c) Let G be the symmetric group S_n, and let H be the subgroup of permutations which fix 1. Describe the double coset decomposition of S_n into double cosets of (H,H).<br />
<br />
==HOMEWORK 2 (due Sep 27)==<br />
<br />
1. Prove that the symmetric group S_n is generated by transpositions. Hint: proceed by induction on n. Let H be the subgroup of S_n consisting of permutations fixing 1. Observe that H is isomorphic to S_{n-1}, so that H is generated by transpositions. Explain why it suffices to show that, for any g in S_n, the coset gH contains a transposition. Then explain why this is the case.<br />
<br />
2. Given that the symmetric group S_n is generated by transpositions, show that there are no nontrivial homomorphisms from S_n to Z/pZ for any odd prime p.<br />
<br />
3. Compute the centralizer of the triple flip (12)(34)(56) in S_6. (Note: this is the kind of problem that a computer algebra package like sage is very good at, and I encourage you to learn to use sage well enough to check your answer.)<br />
<br />
4. Warning: a normal subgroup of a normal subgroup need not be normal! Let H be the subgroup of S_4 generated by (12)(34). Then H is a subgroup of the Klein 4-group V_4, which we have already shown is normal in S_4. Show that H is normal in V_4, but H is not normal in S_4.<br />
<br />
5. The '''quaternion group''' Q is a non-abelian finite group of order 8. Its elements are 1, -1, i, -i, j, -j, k, -k; it satisfies g^2 = -1 for all g except +-1, and ijk = 1. (Note that the notation "-i" means "the product of i with -1" and -1 is understood to be central in Q.) Prove that every subgroup of Q is normal.<br />
<br />
6. (finishing example done in class) Let Gamma be the group F<x,y> / (x^2 = 1, y^2 = 1). (Remember, this means the quotient of the free group by the group generated by all conjugates of x^2 and y^2.) Prove that Gamma is infinite. Prove that Gamma has a subgroup R generated by xy isomorphic to Z, that R is normal, and that the quotient Gamma/R is isomorphic to Z/2Z. <br />
<br />
7. Suppose a group G has the property that every element in g satisfies g^2 = 1. Show that G must be abelian.<br />
<br />
==HOMEWORK 3 (due Oct 4)==<br />
<br />
1. Multiplication by -1 is an automorphism of Z/nZ. Let a be the homomorphism Z/2Z -> Aut(Z/nZ) which sends the nontrivial element of Z/2Z to multiplication by -1. Then we have a semidirect product of Z/pZ by Z/2Z as defined in class. This group of order 2n is called a ''dihedral group.'' and is denoted D_n (or sometimes D_{2n}).<br />
<br />
1a. Compute the center of D_n. (Note that the answer depends on n!)<br />
1b. Show that all involutions of D_n are conjugate to each other if and only if n is odd.<br />
<br />
2. (More on direct products that we didn't do in class.) Suppose that G is a semidirect product of N with H. Suppose furthermore that H is also normal in G. Prove that G is a direct product of N with H.<br />
<br />
3. Let Q be the quaternion group from last week's homework. Show that Q does not decompose as a semidirect product N \rtimes H, apart form the trivial cases N = {e} and N = Q.<br />
<br />
4. The ''affine linear group'' of degree n is the group of transformations from R^n to R^n of the form x -> Ax + b, where A is a matrix in GL_n(R) and b is a vector in R^n. Prove that the affine linear group is a semidirect product R^n \rtimes GL_n.<br />
<br />
5. The ''ordinary triangle group'' T(p,q,r) is the group with presentation <x,y | x^p = y^q = (xy)^r = 1>. This is a very interesting family of groups which arises naturally in many contexts in number theory and topology.<br />
<br />
6a. Prove that T(2,2,n) is isomorphic to the dihedral group D_n.<br />
6b. Prove that T(2,3,3) is finite and compute its order. Is it isomorphic to a finite group we've discussed in class?<br />
<br />
(It is a beautiful fact, but outside the scope of this course, that T(p,q,r) is finite if and only if 1/p + 1/q + 1/r > 1.)<br />
<br />
7. Let G be a subgroup of GL_n(R) which contains SL_n(R). Prove that G is a normal subgroup of GL_n(R).<br />
<br />
8. Let G be a group and let H_1 and H_2 be subgroups of G of finite index (that is, the number of cosets in G/H_i is finite, i = 1,2.) Prove that H_1 intersect H_2 also has finite index in G.<br />
<br />
9. Recall that the commutator [x,y] is x y x^{-1} y^{-1}. If G is a group, the commutator subgroup of G, denoted G' or [G,G], is the subgroup of G generated by all commutators [x,y] for x,y in G.<br />
<br />
9a. Show that G' is a normal subgroup of G.<br />
9b. Show that G/G' is an abelian group.<br />
9c. Show that if f: G -> A is a homomorphism, with A abelian, then G' is contained in ker f. Conclude that f factors through a homomorphism G/G' -> A.<br />
9d. Show that G' = G if and only if G has no nontrivial abelian quotient. In this case, we say G is ''perfect''.<br />
<br />
10. Under what conditions on (p,q,r) is the triangle group T(p,q,r) perfect?<br />
<br />
==HOMEWORK 4 (due Oct 16)==<br />
<br />
1. Let X be a set with a transitive action of a group G, which we think of as a homomorphism f: G -> Sym(X). Let H_x be the stabilizer of an element x of X.<br />
<br />
1a. If x' is another element of X, show that H_{x"} and H_x are conjugate subgroups. (Hint: this is not true without the hypothesis of transitivity, so make sure to use it!)<br />
<br />
1b. Show that the kernel of f is the intersection of all the conjugates of H_x in G. (This subgroup is called the ''normal core'' of H_x.)<br />
<br />
2. This week, a certified mathematical grown-up asked on MathOverflow what the index-3 subgroups of SL_2(Z/pZ) were. Let's do it ourselves for homework. Let p be a prime, and suppose H is an index-3 subgroup of SL_2(Z/pZ). I do not claim that the proof sketched below is necessarily the easiest, but it allows us to cover some ground that we won't get to do in lecture.<br />
<br />
2a. Using the coset action, show that the normal core of H is the kernel of a homomorphism f from SL_2(Z/pZ) to S_3.<br />
<br />
2b. A ''unipotent matrix'' is a matrix A such that (A-I)^n = 0 for some positive integer n. Prove that a unipotent element of SL_2(Z/pZ) has order dividing p.<br />
<br />
2c. You may use the fact that unipotent matrices generate SL_2(Z/pZ). Given this fact, show that f must be trivial when p > 3.<br />
<br />
2d. Using the above, show that SL_2(Z/pZ) has no index-3 subgroups when p > 3.<br />
<br />
2e. (extra credit) What can you say about the minimal index of a proper subgroup of SL_2(Z/pZ)?<br />
<br />
3. Let H be a subgroup of G of index 2. Prove that H is normal.<br />
<br />
4. We proved two theorems in class: Lagrange's theorem, which says that the order of a subgroup of G divides |G|, and Cauchy's theorem, which says that if p divides |G| then there exists a subgroup of G of order p. Cauchy's theorem is a kind of partial converse to Lagrange's theorem, but the full converse is false. Give an example of a finite group G and a divisor n of |G| such that you can prove there is no subgroup of G of order n.<br />
<br />
5. Let X be the set of lines in F_3^2 (here a line means a 1-dimensional linear subspace) so that |X| = 4. Let G be the projective linear group PGL_2(F_3), so that G acts on X via its action on F_3^2. This action can be thought of as a homomorphism from G to S_4. Show that this homomorphism is an ISOMORPHISM from PGL_2(F_3) to S_4.<br />
<br />
6. The group S_3 has only three conjugacy classes. Prove that a finite group with at most three conjugacy classes has order at most 6.<br />
<br />
7. Let X be the set of ordered triples of elements of {1,..,n}, for some n >= 3. Then S_n acts on X. How many orbits are there? (Hint: the answer does not depend on n.)<br />
<br />
8. Suppose that G and H are groups and f: G -> H is a homomorphism. Recall the definition of the abelianization G^ab from the previous problem set.<br />
<br />
8a. Show that the composition G -> H -> H^ab factors through a unique homomorphism G^ab -> H^ab, which we denote f^ab.<br />
<br />
8b. Show that if f: G -> H and g: H -> Q are homomorphisms, then f^ab o g^ab = (f o g)^ab.<br />
<br />
(For those reading MacLane, this constitutes a proof that abelianization is a '''functor''' from the category of groups to the category of abelian groups.)<br />
<br />
==HOMEWORK 5 (due Oct 23)==<br />
<br />
1. Show that there are only two different isomorphism classes of groups of order 14. (Hint: prove that such a group must be a semidirect product of a group of order 7 by a group of order 2.)<br />
<br />
2. (variant of Burnside's Lemma) Let G be a finite group and H a normal subgroup. Suppose G acts on a finite set X. <br />
<br />
2a. Show that if x and y lie in the same H-orbit, and g is an element of G, then gx and gy also lie in the same H-orbit. Thus, G/H has a natural action on the set of H-orbits X/H.<br />
<br />
2b. Show that the average number of fixed points in X of an element of the coset gH is the same as the number of fixed points of the element gH of G/H in its action on X/H.<br />
<br />
3. A ''central extension'' of a group G by an abelian group A is a group E, together with a surjective homomorphism E -> G whose kernel is central in E and is isomorphic to A.<br />
<br />
3a. Show that the quaternion group is a central extension of (Z/2Z)^2 by Z/2Z.<br />
<br />
3b. Give an example of another nonabelian group which is a central extension of (Z/2Z)^2 by Z/2Z and which is not isomorphic to the quaternion group.<br />
<br />
4. Prove that a central extension of an abelian group is nilpotent.<br />
<br />
5. Give two different composition series for S_4 and show that they have the same composition factors.<br />
<br />
6. Suppose that A is a finitely generated abelian group, and B is a subgroup of A.<br />
Prove that rank(A/B) = rank(A) - rank(B).<br />
ADDED: I asserted in class, but did not prove, that a subgroup of a finitely generated abelian group is again finitely generated. As one class member pointed out, without this fact it's not even clear why it makes sense to talk about rank(B)! You may assume it (though it would have been better for me to have included proving this fact as part of the problem....)<br />
<br />
7. For this problem, you may use the fact (which I should have asked on homework earlier) that if H_1 and H_2 are subgroups of G, then<br />
<br />
[G: H_1 intersect H_2] <= [G:H_1][G:H_2].<br />
<br />
Prove that if G is a simple group, d is an integer greater than 1, and S is a subgroup of index d in G, then |G| <= d^d.<br />
<br />
Conclude that, in particular, A_5 has no subgroup of index 3. On the other hand, exhibit a subgroup of A_5 of index 5.<br />
<br />
(Note: d^d is not a sharp bound; can you get a bound of d factorial instead?)<br />
<br />
<!--(This argument shows that a solvable subgroup of A_n has size at most about (n-1)!, but this is far from sharp; e.g. I found a citation to a 1967 paper of Dixon which shows that a solvable subgroup of A_n has size at most k^n where k is around 2.88.)--><br />
<br />
8. Give an example of an abelian group A with a subgroup B such that B is not a direct summand of A.<br />
<br />
9. Let A be the subgroup of Z^2 generated by the vector (1,0). Let B be the subgroup of Z^2 generated by the vector (3,3). Show that A and B are isomorphic to each other, but Z^2 / A is not isomorphic to Z^2 / B.<br />
<br />
==HOMEWORK 6 (due Oct 30)==<br />
<br />
1. In this problem, you will construct a functor F from the category of finite sets to the category of complex vector spaces. I'll tell you what it does on objects: if X is a finite set, then F(X) is the vector space spanned by a set of basis elements {e_x}_{x in X}, so that dim F(X) = |X|. Your job: complete the definition of F by describing the map from Mor_{Finite Sets}(X,Y) to Mor_{Vector Spaces}(F(X),F(Y)), and showing that your map respects composition.<br />
<br />
2. If (V, rho) is a representation of a group G, denote by V^G the space of vectors in V such that gv = v for all g in G; these are called _invariant_ vectors. We saw that the permutation representation of S_3 has a one-dimensional space of invariants. Prove that if X is a finite set with G-action, and V_X the corresponding permutation representation of G, then<br />
<br />
dim V_X^G = number of orbits of X.<br />
<br />
3. (Invariants are functorial) Let G be a group. Then there is a category called C[G]-Mod whose objects are complex representations of G (i.e. pairs (V,rho) where V is a complex vector space and rho is a homomorphism from G to GL(V)). The morphisms in this category are the ones described in class: Mor((V,rho),(W,psi)) is the set of linear maps f from V to W such that<br />
<br />
f(rho(g)(v)) = psi(g)(f(v))<br />
<br />
for all g in G and all v in V.<br />
<br />
Show that there is a functor H_0 from the category of representations of G to the category of vector spaces whose action on objects is given by<br />
<br />
H_0((V) = V^G.<br />
<br />
(In other words: your job again is to explain how to associate to a morphism from (V,rho) to (W,phi) a morphism from V^G to W^G, in a way that's compatible with composition.)<br />
<br />
<!--4. The group PSL_2(Z) has a presentation of the form [U,V|U^2 = V^3 = 1]. Thus, an n-dimensional representation of PSL_2(Z) is precisely a pair of n x n matrices (rho(U),rho(V)), such that rho(U)^2 = rho(V)^3 = 1. Prove that for every n > 1 there are uncountably many isomorphism classes of n-dimensional representations of PSL_2(Z). (Extra credit: show that there are uncountably many isomorphism classes of _irreducible_ representations.)<br />
<br />
An amazing theorem of Margulis (superrigidity) shows that SL_2(Z) is very special among arithmetic groups in this way. For instance, there are only _finitely_ many isomorphism classes of n-dimensional representations of PSL_d(Z) for any d > 2; those groups are "rigid."--><br />
<br />
5. Let G be a finite group, let X be a set with G-action, and let chi be the character of the corresponding permutation representation. Show that, for each g in G, chi(g) is the number of elements of X fixed by g. (I am actually going to prove this in class.)<br />
<br />
6. Let V be the space of homogeneous degree-3 polynomials in variables x_1, x_2, x_3; then S_3 acts on V by permutation of the three variables. Describe the decomposition of V into irreducible representations of S_3.<br />
<br />
7a. Show that if chi is the character of a representation of a finite group G, then chi(g^{-1}) is the complex conjugate of chi.<br />
<br />
7b. As a consequence, show that if g is conjugate to g^{-1} for all g in G, then chi(g) is real.<br />
<br />
7c. The groups D_p and S_n have the property that g is conjugate to g^{-1} for _every_ g in G, so that every character is a real-valued function on G. Give an example of a group G where this is not the case, and give an example of a representation of G whose character takes non-real values.<br />
<br />
==HOMEWORK 7 (due Nov 6)==<br />
<br />
1. Let G be a finite group, let X be a finite set with G-action, and let chi be the character of the corresponding permutation representation. Show that, for each g in G, chi(g) is the number of elements of X fixed by g. (I am actually going to prove this in class.)<br />
<br />
2a. Show that if chi is the character of a finite-dimensional representation of a finite group G, then chi(g^{-1}) is the complex conjugate of chi(g) for all g.<br />
<br />
2b. As a consequence, show that if g is conjugate to g^{-1} for all g in G, then chi(g) is real.<br />
<br />
2c. The groups D_p and S_n have the property that g is conjugate to g^{-1} for _every_ g in G, so that every character is a real-valued function on G. Give an example of a group G where this is not the case, and give an example of a representation of G whose character takes non-real values.<br />
<br />
3. Let V and W be complex vector spaces. Let f be a linear transformation of V, and g a linear transformation of W. <br />
<br />
3a. Show that there is a unique linear transformation F satisfying<br />
<br />
F(v tensor w) = f(v) tensor g(w)<br />
<br />
for all v in V and all w in W. We denote this transformation by f tensor g.<br />
<br />
3b. Suppose V and W are finite dimensional. Show that Trace(f tensor g) = Trace(f) Trace(g). What does this say when f and g are both the identity transformation?<br />
<br />
4. Let V be a vector space. We define Sym^2 V to be the quotient of V tensor V by the subspace generated by all elements of the form<br />
<br />
v tensor w - w tensor v<br />
<br />
for v,w in V.<br />
<br />
Suppose dim V = n. What is dim Sym^2 V?<br />
<br />
5. Show that Z/pZ tensor Z/qZ is zero when p and q are distinct primes.<br />
<br />
==HOMEWORK 8 (due Nov 13)==<br />
<br />
1. (Schur's lemma can be false over non-algebraically closed fields) Let G be Z/3Z and let g be a generator of G. Then G acts by cyclic permutations on the set {1,2,3}; let P be the corresponding permutation representation on the REAL (not complex!) vector space generated by e_1,e_2,e_3.<br />
<br />
1a. Show that P breaks up as the direct sum of the trivial representation and a 2-dimensional representation V which is irreducible. (Remember, we are not working over the complex numbers, so you cannot use the criterion for irreducibility in terms of the character that we worked out in class for the complex case; you need to prove directly that V has no nontrivial proper subrepresentation.)<br />
<br />
1b. Show that the map from V to V induced by g is G-equivariant, but is neither zero nor a scalar multiplication.<br />
<br />
2. If V_1 and V_2 are representations (not necessarily irreducible!) of G, show that <chi_V_1, chi V_2> is a non-negative integer.<br />
<br />
3. When a formula always yields a non-negative integer, it is often because it is secretly a formula for either the cardinality of some finite set or the dimension of some finite-dimensional vector space. In this case, show that<br />
<br />
<chi_{V_1}, chi_{V_2}> = dim_C Hom_G(V_1,V_2)<br />
<br />
where we recall that Hom_G(V_1,G_2) refers to the space of G-equivariant homomorphisms from V_1 to V_2.<br />
<br />
4. If A is an abelian group, show that every irreducible representation of A is 1-dimensional. (Hint: show that there is a vector which is an eigenvector for every element of A!)<br />
<br />
5. Let G be the group of order 20 which is the semidirect product of A by H, where A is Z/5Z and H is (Z/5Z)^* with its natural action on A. We will work out all the irreducible representations of A. <br />
<br />
5a. First of all, consider the 5-dimensional permutation representation of G acting by left multiplication on the cosets of H. Show that this representation decomposes as a direct sum of the trivial representation and an irreducible 4-dimensional representation. (Hint: it is probably easier to use the character of this representation than to prove irreducibility directly.) <br />
<br />
5b. Now show that there are 4 1-dimensional representations of G which factor through the quotient G/A (i.e. such that rho(A) is the identity.)<br />
<br />
5c. Using the decomposition of the regular representation, show that the irreducible representations of G you have constructed are the only ones.<br />
<br />
==HOMEWORK 9 (due Nov 20)==<br />
<br />
1. If R is a ring (not necessarily with 1) such that a^2 = a for every a in R, R is called a Boolean ring.<br />
<br />
1a. Show that every Boolean ring is commutative.<br />
<br />
1b. Show that any nonzero Boolean ring has characteristic 2. (In other words, show that a+a=0 for all a in R.)<br />
<br />
<br />
2. Let phi : R -> S be a homomorphism of commutative rings with 1.<br />
<br />
2a. If P is a prime ideal of S, show that phi^(-1) (P) is either R or a prime ideal of R.<br />
<br />
2b. If M is a maximal ideal of S and phi is surjective, show that phi^(-1) (M) is a maximal ideal of R.<br />
<br />
2c. Find a homomorphism phi : R -> S and a maximal ideal M of S such that phi^(-1) (M) is not a maximal ideal of R.<br />
<br />
<br />
3. Let Nil(A) denote the set of nilpotent elements of a ring A, and let R be a commutative ring. In class it was observed that Nil(R) is an ideal of R.<br />
<br />
3a. Find (a generator of) Nil(Z/720Z).<br />
<br />
3b. Show that Nil(R/Nil(R)) = 0.<br />
<br />
3c. If x is in Nil(R) and u is a unit, show that u+x is a unit.<br />
<br />
3d. Show that Nil(S), where S is the 2x2 matrices with real coefficients, is not an ideal of S.<br />
<br />
<br />
4a. If M is an R-module and I is an ideal of R such that im = 0 for all i in I and m in M, show that M carries the structure of an R/I-module.<br />
<br />
4b. If S is a commutative ring with 1 and J is a maximal ideal of S, show that J/J^2 is a vector space over S/J.<br />
<br />
<br />
5. A "divisible element" x of a Z-module is one such that for each positive integer k, there exists a y such that ky=x. Show that a free Z-module cannot contain a divisible element.<br />
<br />
<br />
6. Prove that the direct sum of any collection of free R-modules is also a free R-module.<br />
<br />
<br />
7. In this problem you will prove that the module M given by the direct product of copies of Z, indexed by the positive integers, is not a free Z-module. Therefore, assume that M is a free Z-module with basis B, and let N be the direct sum of copies of Z, as a submodule of M.<br />
<br />
7a. Show there is a countable subset B1 of B such that N is contained in the submodule N1 generated by B1. [Hint: N is countable.]<br />
<br />
7b. Show that M/N1 is a free Z-module.<br />
<br />
7c. Let S = {(b1, b2, b3, ...) : b_i = i! or -i!}. Show that there is some s in S which is not in N1. [Hint: S is uncountable.]<br />
<br />
7d. For any s in S but not N1, show that for each positive integer k, there exists some m in M with s-km in N1. Conclude that s+N1 is a divisible element of M/N1.<br />
<br />
7e. Use problem 5 to obtain a contradiction.<br />
<br />
<br />
<br />
8. In class I (will have) said "rank" is not always well-defined for R-modules if R is not commutative. Here is an example demonstrating this assertion. Let M be the direct product of copies of Z as in exercise 7, and let R be the endomorphism ring of M. Define phi_1(a1,a2,a3,a4,...) = (a1,a3,a5,...) and phi_2(a1,a2,a3,a4,...) = (a2,a4,a6,...).<br />
<br />
8a. Show that {phi_1, phi_2} is a free basis for R, as a left R-module. [Hint: let psi_1(a1,a2,a3,...) = (a1,0,a2,0,...) and psi_2(a1,a2,a3,...) = (0,a1,0,a2,...). Compute the products of the phi_i and psi_j in either order, and use the result to show that the phi_i are a basis.]<br />
<br />
8b. Show that R is isomorphic to R^n for every positive integer n.</div>Dummithttps://hilbert.math.wisc.edu/wiki/index.php?title=741&diff=47027412012-11-13T19:11:31Z<p>Dummit: Homework 9</p>
<hr />
<div>'''Math 741'''<br />
<br />
Algebra<br />
<br />
Prof: [http://www.math.wisc.edu/~ellenber Jordan Ellenberg]<br />
<br />
Grader: Evan Dummit<br />
<br />
Ellenberg's office hours: Friday 3pm.<br />
<br />
Grader's office hours: Monday 4pm. Late homework may be given directly to the grader, along with either (i) the instructor's permission, or (ii) a polite request for mercy.<br />
<br />
This course, the first semester of the introductory graduate sequence in algebra, will cover the basic theory of groups, group actions, representation, linear and multilinear algebra, and the beginnings of ring theory.<br />
<br />
==SYLLABUS==<br />
<br />
In this space we will record the theorems and definitions we covered each week, which we can use as a list of notions you should be prepared to answer questions about on the Algebra qualifying exam. The material covered on the homework is also an excellent guide to the scope of the course. <br />
<br />
'''WEEK 1''': <br />
<br />
Definition of group. Associativity. Inverse. <br />
<br />
Examples of group: GL_n(R). GL_n(Z). Z/nZ. R. Z. R^*. The free group F_k on k generators. <br />
<br />
Homomorphisms. The homomorphisms from F_k to G are in bijection with G^k. Isomorphisms.<br />
<br />
'''WEEK 2''':<br />
<br />
The symmetric group (or permutation group) S_n on n letters. Cycle decomposition of a permutation. Order of a permutation. Thm: every element of a finite group has finite order. <br />
<br />
Subgroups. Left and right cosets. Lagrange's Theorem. Cyclic groups. The order of an element of a finite group is a divisor of the order of the group.<br />
<br />
The sign homomorphism S_n -> +-1.<br />
<br />
<br />
'''WEEK 3'''<br />
<br />
Normal subgroups. The quotient of a group by a normal subgroup. The first isomorphism theorem. Examples of S_n -> +-1 and S_4 -> S_3 with kernel V_4, the Klein 4-group.<br />
<br />
Centralizers and centers. Abelian groups. The center of SL_n(R) is either 1 or +-1.<br />
<br />
Groups with presentations. The infinite dihedral group <x,y | x^2 = 1, y^2 = 1>.<br />
<br />
'''WEEK 4'''<br />
<br />
More on groups with presentations.<br />
<br />
Second and third isomorphism theorems.<br />
<br />
Semidirect products.<br />
<br />
'''WEEK 5'''<br />
<br />
Group actions, orbits, and stabilizers.<br />
<br />
Orbit-stabilizer theorem.<br />
<br />
Cayley's theorem.<br />
<br />
Cauchy's theorem.<br />
<br />
'''WEEK 6'''<br />
<br />
Applications of orbit-stabilizer theorem (p-groups have nontrivial center, first Sylow theorem.)<br />
<br />
Classification of finite abelian groups and finitely generated abelian groups.<br />
<br />
Composition series and the Jordan-Holder theorem (which we state but don't prove.) <br />
<br />
The difference between knowing the composition factors and knowing the group (e.g. all p-groups of the same order have the same composition factors.)<br />
<br />
'''WEEK 7'''<br />
<br />
Simplicity of A_n.<br />
<br />
Nilpotent groups (main example: the Heisenberg group)<br />
<br />
Derived series and lower central series.<br />
<br />
Category theory 101: Definition of category and functor. Some examples. A group is a groupoid with one object.<br />
<br />
'''WEEK 8'''<br />
<br />
Introduction to representation theory. <br />
<br />
<br />
<br />
Below you will find a repository of homework problems. Note that some of these problems are taken from Lang's ''Algebra''.<br />
<br />
==HOMEWORK 1 (due Sep 20)==<br />
<br />
1. Suppose that H_1 and H_2 are subgroups of a group G. Prove that the intersection of H_1 and H_2 is a subgroup of G.<br />
<br />
2. Recall that S_3 (the symmetric group) is the group of permutations of the set {1..3}. List all the subgroups of S_3.<br />
<br />
3. We can define an equivalence relation on rational numbers by declaring two rational numbers to be equal whenever they differ by an integer. We denote the set of equivalence classes by Q/Z. The operation of addition makes Q/Z into a group.<br />
<br />
a) For each n, prove that Q/Z has a subgroup of order n.<br />
<br />
b) Prove that Q/Z is a ''divisible'' group: that is, if x is an element of Q/Z and n is an integer, there exists an element y of Q/Z such that ny = x. (Note that we write the operation in this group as addition rather than multiplication, which is why we write ny for the n-fold product of y with itself rather than y^n)<br />
<br />
c) Prove that Q/Z is not finitely generated. (Hint: prove that if x_1, .. x_d is a finite subset of Q/Z, the subgroup of Q/Z generated by x_1, ... x_d is finite.)<br />
<br />
4. Let F_2 be the free group on two generators x,y. Prove that there is a automorphism of F_2 which sends x to xyx and y to xy, and prove that this automorphism is unique.<br />
<br />
5. Let H be a subgroup of G, and let N_G(H), the ''normalizer'' of H in G, be the set of elements g in G satsifying g H g^{-1} = H. Prove that N_G(H) is a subgroup of G.<br />
<br />
6. Let T be the subgroup of GL_2(R) consisting of diagonal matrices. This is an example of a "Cartan subgroup," or "torus" (whence the notation T.) Describe the normalizer N(T) of T in GL_2(R). Prove that there is a homomorphism from N(T) to Z/2Z whose kernel is precisely T. <br />
<br />
7. Prove that the group of inner automorphisms of a group is normal in the full automorphism group Aut(G).<br />
<br />
8. Let H and H' be subgroups of a group G, and let x be an element of G. We denote by HxH' the set of elements of the form hxh', with h in H and h' in H', and we call HxH a ''double coset'' of the pair (H,H').<br />
<br />
a) Show that G decomposes as a disjoint union of double cosets of (H,H').<br />
<br />
b) Let G be GL_2(R) and let B be the subgroup of upper triangular matrices. Prove that G decomposes into exactly two double cosets of (B,B).<br />
<br />
c) Let G be the symmetric group S_n, and let H be the subgroup of permutations which fix 1. Describe the double coset decomposition of S_n into double cosets of (H,H).<br />
<br />
==HOMEWORK 2 (due Sep 27)==<br />
<br />
1. Prove that the symmetric group S_n is generated by transpositions. Hint: proceed by induction on n. Let H be the subgroup of S_n consisting of permutations fixing 1. Observe that H is isomorphic to S_{n-1}, so that H is generated by transpositions. Explain why it suffices to show that, for any g in S_n, the coset gH contains a transposition. Then explain why this is the case.<br />
<br />
2. Given that the symmetric group S_n is generated by transpositions, show that there are no nontrivial homomorphisms from S_n to Z/pZ for any odd prime p.<br />
<br />
3. Compute the centralizer of the triple flip (12)(34)(56) in S_6. (Note: this is the kind of problem that a computer algebra package like sage is very good at, and I encourage you to learn to use sage well enough to check your answer.)<br />
<br />
4. Warning: a normal subgroup of a normal subgroup need not be normal! Let H be the subgroup of S_4 generated by (12)(34). Then H is a subgroup of the Klein 4-group V_4, which we have already shown is normal in S_4. Show that H is normal in V_4, but H is not normal in S_4.<br />
<br />
5. The '''quaternion group''' Q is a non-abelian finite group of order 8. Its elements are 1, -1, i, -i, j, -j, k, -k; it satisfies g^2 = -1 for all g except +-1, and ijk = 1. (Note that the notation "-i" means "the product of i with -1" and -1 is understood to be central in Q.) Prove that every subgroup of Q is normal.<br />
<br />
6. (finishing example done in class) Let Gamma be the group F<x,y> / (x^2 = 1, y^2 = 1). (Remember, this means the quotient of the free group by the group generated by all conjugates of x^2 and y^2.) Prove that Gamma is infinite. Prove that Gamma has a subgroup R generated by xy isomorphic to Z, that R is normal, and that the quotient Gamma/R is isomorphic to Z/2Z. <br />
<br />
7. Suppose a group G has the property that every element in g satisfies g^2 = 1. Show that G must be abelian.<br />
<br />
==HOMEWORK 3 (due Oct 4)==<br />
<br />
1. Multiplication by -1 is an automorphism of Z/nZ. Let a be the homomorphism Z/2Z -> Aut(Z/nZ) which sends the nontrivial element of Z/2Z to multiplication by -1. Then we have a semidirect product of Z/pZ by Z/2Z as defined in class. This group of order 2n is called a ''dihedral group.'' and is denoted D_n (or sometimes D_{2n}).<br />
<br />
1a. Compute the center of D_n. (Note that the answer depends on n!)<br />
1b. Show that all involutions of D_n are conjugate to each other if and only if n is odd.<br />
<br />
2. (More on direct products that we didn't do in class.) Suppose that G is a semidirect product of N with H. Suppose furthermore that H is also normal in G. Prove that G is a direct product of N with H.<br />
<br />
3. Let Q be the quaternion group from last week's homework. Show that Q does not decompose as a semidirect product N \rtimes H, apart form the trivial cases N = {e} and N = Q.<br />
<br />
4. The ''affine linear group'' of degree n is the group of transformations from R^n to R^n of the form x -> Ax + b, where A is a matrix in GL_n(R) and b is a vector in R^n. Prove that the affine linear group is a semidirect product R^n \rtimes GL_n.<br />
<br />
5. The ''ordinary triangle group'' T(p,q,r) is the group with presentation <x,y | x^p = y^q = (xy)^r = 1>. This is a very interesting family of groups which arises naturally in many contexts in number theory and topology.<br />
<br />
6a. Prove that T(2,2,n) is isomorphic to the dihedral group D_n.<br />
6b. Prove that T(2,3,3) is finite and compute its order. Is it isomorphic to a finite group we've discussed in class?<br />
<br />
(It is a beautiful fact, but outside the scope of this course, that T(p,q,r) is finite if and only if 1/p + 1/q + 1/r > 1.)<br />
<br />
7. Let G be a subgroup of GL_n(R) which contains SL_n(R). Prove that G is a normal subgroup of GL_n(R).<br />
<br />
8. Let G be a group and let H_1 and H_2 be subgroups of G of finite index (that is, the number of cosets in G/H_i is finite, i = 1,2.) Prove that H_1 intersect H_2 also has finite index in G.<br />
<br />
9. Recall that the commutator [x,y] is x y x^{-1} y^{-1}. If G is a group, the commutator subgroup of G, denoted G' or [G,G], is the subgroup of G generated by all commutators [x,y] for x,y in G.<br />
<br />
9a. Show that G' is a normal subgroup of G.<br />
9b. Show that G/G' is an abelian group.<br />
9c. Show that if f: G -> A is a homomorphism, with A abelian, then G' is contained in ker f. Conclude that f factors through a homomorphism G/G' -> A.<br />
9d. Show that G' = G if and only if G has no nontrivial abelian quotient. In this case, we say G is ''perfect''.<br />
<br />
10. Under what conditions on (p,q,r) is the triangle group T(p,q,r) perfect?<br />
<br />
==HOMEWORK 4 (due Oct 16)==<br />
<br />
1. Let X be a set with a transitive action of a group G, which we think of as a homomorphism f: G -> Sym(X). Let H_x be the stabilizer of an element x of X.<br />
<br />
1a. If x' is another element of X, show that H_{x"} and H_x are conjugate subgroups. (Hint: this is not true without the hypothesis of transitivity, so make sure to use it!)<br />
<br />
1b. Show that the kernel of f is the intersection of all the conjugates of H_x in G. (This subgroup is called the ''normal core'' of H_x.)<br />
<br />
2. This week, a certified mathematical grown-up asked on MathOverflow what the index-3 subgroups of SL_2(Z/pZ) were. Let's do it ourselves for homework. Let p be a prime, and suppose H is an index-3 subgroup of SL_2(Z/pZ). I do not claim that the proof sketched below is necessarily the easiest, but it allows us to cover some ground that we won't get to do in lecture.<br />
<br />
2a. Using the coset action, show that the normal core of H is the kernel of a homomorphism f from SL_2(Z/pZ) to S_3.<br />
<br />
2b. A ''unipotent matrix'' is a matrix A such that (A-I)^n = 0 for some positive integer n. Prove that a unipotent element of SL_2(Z/pZ) has order dividing p.<br />
<br />
2c. You may use the fact that unipotent matrices generate SL_2(Z/pZ). Given this fact, show that f must be trivial when p > 3.<br />
<br />
2d. Using the above, show that SL_2(Z/pZ) has no index-3 subgroups when p > 3.<br />
<br />
2e. (extra credit) What can you say about the minimal index of a proper subgroup of SL_2(Z/pZ)?<br />
<br />
3. Let H be a subgroup of G of index 2. Prove that H is normal.<br />
<br />
4. We proved two theorems in class: Lagrange's theorem, which says that the order of a subgroup of G divides |G|, and Cauchy's theorem, which says that if p divides |G| then there exists a subgroup of G of order p. Cauchy's theorem is a kind of partial converse to Lagrange's theorem, but the full converse is false. Give an example of a finite group G and a divisor n of |G| such that you can prove there is no subgroup of G of order n.<br />
<br />
5. Let X be the set of lines in F_3^2 (here a line means a 1-dimensional linear subspace) so that |X| = 4. Let G be the projective linear group PGL_2(F_3), so that G acts on X via its action on F_3^2. This action can be thought of as a homomorphism from G to S_4. Show that this homomorphism is an ISOMORPHISM from PGL_2(F_3) to S_4.<br />
<br />
6. The group S_3 has only three conjugacy classes. Prove that a finite group with at most three conjugacy classes has order at most 6.<br />
<br />
7. Let X be the set of ordered triples of elements of {1,..,n}, for some n >= 3. Then S_n acts on X. How many orbits are there? (Hint: the answer does not depend on n.)<br />
<br />
8. Suppose that G and H are groups and f: G -> H is a homomorphism. Recall the definition of the abelianization G^ab from the previous problem set.<br />
<br />
8a. Show that the composition G -> H -> H^ab factors through a unique homomorphism G^ab -> H^ab, which we denote f^ab.<br />
<br />
8b. Show that if f: G -> H and g: H -> Q are homomorphisms, then f^ab o g^ab = (f o g)^ab.<br />
<br />
(For those reading MacLane, this constitutes a proof that abelianization is a '''functor''' from the category of groups to the category of abelian groups.)<br />
<br />
==HOMEWORK 5 (due Oct 23)==<br />
<br />
1. Show that there are only two different isomorphism classes of groups of order 14. (Hint: prove that such a group must be a semidirect product of a group of order 7 by a group of order 2.)<br />
<br />
2. (variant of Burnside's Lemma) Let G be a finite group and H a normal subgroup. Suppose G acts on a finite set X. <br />
<br />
2a. Show that if x and y lie in the same H-orbit, and g is an element of G, then gx and gy also lie in the same H-orbit. Thus, G/H has a natural action on the set of H-orbits X/H.<br />
<br />
2b. Show that the average number of fixed points in X of an element of the coset gH is the same as the number of fixed points of the element gH of G/H in its action on X/H.<br />
<br />
3. A ''central extension'' of a group G by an abelian group A is a group E, together with a surjective homomorphism E -> G whose kernel is central in E and is isomorphic to A.<br />
<br />
3a. Show that the quaternion group is a central extension of (Z/2Z)^2 by Z/2Z.<br />
<br />
3b. Give an example of another nonabelian group which is a central extension of (Z/2Z)^2 by Z/2Z and which is not isomorphic to the quaternion group.<br />
<br />
4. Prove that a central extension of an abelian group is nilpotent.<br />
<br />
5. Give two different composition series for S_4 and show that they have the same composition factors.<br />
<br />
6. Suppose that A is a finitely generated abelian group, and B is a subgroup of A.<br />
Prove that rank(A/B) = rank(A) - rank(B).<br />
ADDED: I asserted in class, but did not prove, that a subgroup of a finitely generated abelian group is again finitely generated. As one class member pointed out, without this fact it's not even clear why it makes sense to talk about rank(B)! You may assume it (though it would have been better for me to have included proving this fact as part of the problem....)<br />
<br />
7. For this problem, you may use the fact (which I should have asked on homework earlier) that if H_1 and H_2 are subgroups of G, then<br />
<br />
[G: H_1 intersect H_2] <= [G:H_1][G:H_2].<br />
<br />
Prove that if G is a simple group, d is an integer greater than 1, and S is a subgroup of index d in G, then |G| <= d^d.<br />
<br />
Conclude that, in particular, A_5 has no subgroup of index 3. On the other hand, exhibit a subgroup of A_5 of index 5.<br />
<br />
(Note: d^d is not a sharp bound; can you get a bound of d factorial instead?)<br />
<br />
<!--(This argument shows that a solvable subgroup of A_n has size at most about (n-1)!, but this is far from sharp; e.g. I found a citation to a 1967 paper of Dixon which shows that a solvable subgroup of A_n has size at most k^n where k is around 2.88.)--><br />
<br />
8. Give an example of an abelian group A with a subgroup B such that B is not a direct summand of A.<br />
<br />
9. Let A be the subgroup of Z^2 generated by the vector (1,0). Let B be the subgroup of Z^2 generated by the vector (3,3). Show that A and B are isomorphic to each other, but Z^2 / A is not isomorphic to Z^2 / B.<br />
<br />
==HOMEWORK 6 (due Oct 30)==<br />
<br />
1. In this problem, you will construct a functor F from the category of finite sets to the category of complex vector spaces. I'll tell you what it does on objects: if X is a finite set, then F(X) is the vector space spanned by a set of basis elements {e_x}_{x in X}, so that dim F(X) = |X|. Your job: complete the definition of F by describing the map from Mor_{Finite Sets}(X,Y) to Mor_{Vector Spaces}(F(X),F(Y)), and showing that your map respects composition.<br />
<br />
2. If (V, rho) is a representation of a group G, denote by V^G the space of vectors in V such that gv = v for all g in G; these are called _invariant_ vectors. We saw that the permutation representation of S_3 has a one-dimensional space of invariants. Prove that if X is a finite set with G-action, and V_X the corresponding permutation representation of G, then<br />
<br />
dim V_X^G = number of orbits of X.<br />
<br />
3. (Invariants are functorial) Let G be a group. Then there is a category called C[G]-Mod whose objects are complex representations of G (i.e. pairs (V,rho) where V is a complex vector space and rho is a homomorphism from G to GL(V)). The morphisms in this category are the ones described in class: Mor((V,rho),(W,psi)) is the set of linear maps f from V to W such that<br />
<br />
f(rho(g)(v)) = psi(g)(f(v))<br />
<br />
for all g in G and all v in V.<br />
<br />
Show that there is a functor H_0 from the category of representations of G to the category of vector spaces whose action on objects is given by<br />
<br />
H_0((V) = V^G.<br />
<br />
(In other words: your job again is to explain how to associate to a morphism from (V,rho) to (W,phi) a morphism from V^G to W^G, in a way that's compatible with composition.)<br />
<br />
<!--4. The group PSL_2(Z) has a presentation of the form [U,V|U^2 = V^3 = 1]. Thus, an n-dimensional representation of PSL_2(Z) is precisely a pair of n x n matrices (rho(U),rho(V)), such that rho(U)^2 = rho(V)^3 = 1. Prove that for every n > 1 there are uncountably many isomorphism classes of n-dimensional representations of PSL_2(Z). (Extra credit: show that there are uncountably many isomorphism classes of _irreducible_ representations.)<br />
<br />
An amazing theorem of Margulis (superrigidity) shows that SL_2(Z) is very special among arithmetic groups in this way. For instance, there are only _finitely_ many isomorphism classes of n-dimensional representations of PSL_d(Z) for any d > 2; those groups are "rigid."--><br />
<br />
5. Let G be a finite group, let X be a set with G-action, and let chi be the character of the corresponding permutation representation. Show that, for each g in G, chi(g) is the number of elements of X fixed by g. (I am actually going to prove this in class.)<br />
<br />
6. Let V be the space of homogeneous degree-3 polynomials in variables x_1, x_2, x_3; then S_3 acts on V by permutation of the three variables. Describe the decomposition of V into irreducible representations of S_3.<br />
<br />
7a. Show that if chi is the character of a representation of a finite group G, then chi(g^{-1}) is the complex conjugate of chi.<br />
<br />
7b. As a consequence, show that if g is conjugate to g^{-1} for all g in G, then chi(g) is real.<br />
<br />
7c. The groups D_p and S_n have the property that g is conjugate to g^{-1} for _every_ g in G, so that every character is a real-valued function on G. Give an example of a group G where this is not the case, and give an example of a representation of G whose character takes non-real values.<br />
<br />
==HOMEWORK 7 (due Nov 6)==<br />
<br />
1. Let G be a finite group, let X be a finite set with G-action, and let chi be the character of the corresponding permutation representation. Show that, for each g in G, chi(g) is the number of elements of X fixed by g. (I am actually going to prove this in class.)<br />
<br />
2a. Show that if chi is the character of a finite-dimensional representation of a finite group G, then chi(g^{-1}) is the complex conjugate of chi(g) for all g.<br />
<br />
2b. As a consequence, show that if g is conjugate to g^{-1} for all g in G, then chi(g) is real.<br />
<br />
2c. The groups D_p and S_n have the property that g is conjugate to g^{-1} for _every_ g in G, so that every character is a real-valued function on G. Give an example of a group G where this is not the case, and give an example of a representation of G whose character takes non-real values.<br />
<br />
3. Let V and W be complex vector spaces. Let f be a linear transformation of V, and g a linear transformation of W. <br />
<br />
3a. Show that there is a unique linear transformation F satisfying<br />
<br />
F(v tensor w) = f(v) tensor g(w)<br />
<br />
for all v in V and all w in W. We denote this transformation by f tensor g.<br />
<br />
3b. Suppose V and W are finite dimensional. Show that Trace(f tensor g) = Trace(f) Trace(g). What does this say when f and g are both the identity transformation?<br />
<br />
4. Let V be a vector space. We define Sym^2 V to be the quotient of V tensor V by the subspace generated by all elements of the form<br />
<br />
v tensor w - w tensor v<br />
<br />
for v,w in V.<br />
<br />
Suppose dim V = n. What is dim Sym^2 V?<br />
<br />
5. Show that Z/pZ tensor Z/qZ is zero when p and q are distinct primes.<br />
<br />
==HOMEWORK 8 (due Nov 13)==<br />
<br />
1. (Schur's lemma can be false over non-algebraically closed fields) Let G be Z/3Z and let g be a generator of G. Then G acts by cyclic permutations on the set {1,2,3}; let P be the corresponding permutation representation on the REAL (not complex!) vector space generated by e_1,e_2,e_3.<br />
<br />
1a. Show that P breaks up as the direct sum of the trivial representation and a 2-dimensional representation V which is irreducible. (Remember, we are not working over the complex numbers, so you cannot use the criterion for irreducibility in terms of the character that we worked out in class for the complex case; you need to prove directly that V has no nontrivial proper subrepresentation.)<br />
<br />
1b. Show that the map from V to V induced by g is G-equivariant, but is neither zero nor a scalar multiplication.<br />
<br />
2. If V_1 and V_2 are representations (not necessarily irreducible!) of G, show that <chi_V_1, chi V_2> is a non-negative integer.<br />
<br />
3. When a formula always yields a non-negative integer, it is often because it is secretly a formula for either the cardinality of some finite set or the dimension of some finite-dimensional vector space. In this case, show that<br />
<br />
<chi_{V_1}, chi_{V_2}> = dim_C Hom_G(V_1,V_2)<br />
<br />
where we recall that Hom_G(V_1,G_2) refers to the space of G-equivariant homomorphisms from V_1 to V_2.<br />
<br />
4. If A is an abelian group, show that every irreducible representation of A is 1-dimensional. (Hint: show that there is a vector which is an eigenvector for every element of A!)<br />
<br />
5. Let G be the group of order 20 which is the semidirect product of A by H, where A is Z/5Z and H is (Z/5Z)^* with its natural action on A. We will work out all the irreducible representations of A. <br />
<br />
5a. First of all, consider the 5-dimensional permutation representation of G acting by left multiplication on the cosets of H. Show that this representation decomposes as a direct sum of the trivial representation and an irreducible 4-dimensional representation. (Hint: it is probably easier to use the character of this representation than to prove irreducibility directly.) <br />
<br />
5b. Now show that there are 4 1-dimensional representations of G which factor through the quotient G/A (i.e. such that rho(A) is the identity.)<br />
<br />
5c. Using the decomposition of the regular representation, show that the irreducible representations of G you have constructed are the only ones.<br />
<br />
==HOMEWORK 9 (due Nov 20)==<br />
<br />
1. If R is a ring (not necessarily with 1) such that a^2 = a for every a in R, R is called a Boolean ring.<br />
<br />
1a. Show that every Boolean ring is commutative.<br />
<br />
1b. Show that any nonzero Boolean ring has characteristic 2. (In other words, show that a+a=0 for all a in R.)<br />
<br />
<br />
2. Let phi : R -> S be a homomorphism of commutative rings with 1.<br />
<br />
2a. If P is a prime ideal of S, show that phi^(-1) (P) is either R or a prime ideal of R.<br />
<br />
2b. If M is a maximal ideal of S and phi is surjective, show that phi^(-1) (M) is a maximal ideal of R.<br />
<br />
2c. Find a homomorphism phi : R -> S and a maximal ideal M of S such that phi^(-1) (M) is not a maximal ideal of R.<br />
<br />
<br />
3. Let Nil(A) denote the set of nilpotent elements of a ring A, and let R be a commutative ring. In class it was observed that Nil(R) is an ideal of R.<br />
<br />
3a. Find (a generator of) Nil(Z/720Z).<br />
<br />
3b. Show that Nil(R/Nil(R)) = 0.<br />
<br />
3c. If x is in Nil(R) and u is a unit, show that u+x is a unit.<br />
<br />
3d. Show that Nil(S), where S is the 2x2 matrices with real coefficients, is not an ideal of S.<br />
<br />
<br />
4a. If M is an R-module and I is an ideal of R such that im = 0 for all i in I and m in M, show that M carries the structure of an R/I-module.<br />
<br />
4b. If S is a commutative ring with 1 and J is a maximal ideal of S, show that J/J^2 is a vector space over S/J.<br />
<br />
<br />
5. A "divisible element" x of a Z-module is one such that for each positive integer k, there exists a y such that ky=x. Show that a free Z-module cannot contain a divisible element.<br />
<br />
<br />
6. Prove that the direct sum of any collection of free R-modules is also a free R-module.<br />
<br />
<br />
7. In this problem you will prove that the module M given by the direct product of copies of Z, indexed by the positive integers, is not a free Z-module. Therefore, assume that M is a free Z-module with basis B, and let N be the direct sum of copies of Z, as a submodule of M.<br />
<br />
7a. Show there is a countable subset B1 of B such that N is contained in the submodule N1 generated by B1. [Hint: N is countable.]<br />
<br />
7b. Show that M/N1 is a free Z-module.<br />
<br />
7c. Let S = {(b1, b2, b3, ...) : b_i = i! or -i!}. Show that there is some s in S which is not in N1. [Hint: S is uncountable.]<br />
<br />
7d. For any s in S but not N1, show that for each positive integer k, there exists some m in M with s-km in N1. Conclude that s+N1 is a divisible element of M/N1.<br />
<br />
7e. Use problem 5 to obtain a contradiction.<br />
<br />
<br />
<br />
8. In class I (will have) said "rank" is not always well-defined for R-modules if R is not commutative. Here is an example demonstrating this assertion. Let M be the direct product of copies of Z as in exercise 6, and let R be the endomorphism ring of M. Define phi_1(a1,a2,a3,a4,...) = (a1,a3,a5,...) and phi_2(a1,a2,a3,a4,...) = (a2,a4,a6,...).<br />
<br />
8a. Show that {phi_1, phi_2} is a free basis for R, as a left R-module. [Hint: let psi_1(a1,a2,a3,...) = (a1,0,a2,0,...) and psi_2(a1,a2,a3,...) = (0,a1,0,a2,...). Compute the products of the phi_i and psi_j in either order, and use the result to show that the phi_i are a basis.]<br />
<br />
8b. Show that R is isomorphic to R^n for every positive integer n.</div>Dummithttps://hilbert.math.wisc.edu/wiki/index.php?title=741&diff=46717412012-11-06T13:42:57Z<p>Dummit: </p>
<hr />
<div>'''Math 741'''<br />
<br />
Algebra<br />
<br />
Prof: [http://www.math.wisc.edu/~ellenber Jordan Ellenberg]<br />
<br />
Grader: Evan Dummit<br />
<br />
Ellenberg's office hours: Friday 3pm.<br />
<br />
Grader's office hours: Monday 4pm. Late homework may be given directly to the grader, along with either (i) the instructor's permission, or (ii) a polite request for mercy.<br />
<br />
This course, the first semester of the introductory graduate sequence in algebra, will cover the basic theory of groups, group actions, representation, linear and multilinear algebra, and the beginnings of ring theory.<br />
<br />
==SYLLABUS==<br />
<br />
In this space we will record the theorems and definitions we covered each week, which we can use as a list of notions you should be prepared to answer questions about on the Algebra qualifying exam. The material covered on the homework is also an excellent guide to the scope of the course. <br />
<br />
'''WEEK 1''': <br />
<br />
Definition of group. Associativity. Inverse. <br />
<br />
Examples of group: GL_n(R). GL_n(Z). Z/nZ. R. Z. R^*. The free group F_k on k generators. <br />
<br />
Homomorphisms. The homomorphisms from F_k to G are in bijection with G^k. Isomorphisms.<br />
<br />
'''WEEK 2''':<br />
<br />
The symmetric group (or permutation group) S_n on n letters. Cycle decomposition of a permutation. Order of a permutation. Thm: every element of a finite group has finite order. <br />
<br />
Subgroups. Left and right cosets. Lagrange's Theorem. Cyclic groups. The order of an element of a finite group is a divisor of the order of the group.<br />
<br />
The sign homomorphism S_n -> +-1.<br />
<br />
<br />
'''WEEK 3'''<br />
<br />
Normal subgroups. The quotient of a group by a normal subgroup. The first isomorphism theorem. Examples of S_n -> +-1 and S_4 -> S_3 with kernel V_4, the Klein 4-group.<br />
<br />
Centralizers and centers. Abelian groups. The center of SL_n(R) is either 1 or +-1.<br />
<br />
Groups with presentations. The infinite dihedral group <x,y | x^2 = 1, y^2 = 1>.<br />
<br />
'''WEEK 4'''<br />
<br />
More on groups with presentations.<br />
<br />
Second and third isomorphism theorems.<br />
<br />
Semidirect products.<br />
<br />
'''WEEK 5'''<br />
<br />
Group actions, orbits, and stabilizers.<br />
<br />
Orbit-stabilizer theorem.<br />
<br />
Cayley's theorem.<br />
<br />
Cauchy's theorem.<br />
<br />
'''WEEK 6'''<br />
<br />
Applications of orbit-stabilizer theorem (p-groups have nontrivial center, first Sylow theorem.)<br />
<br />
Classification of finite abelian groups and finitely generated abelian groups.<br />
<br />
Composition series and the Jordan-Holder theorem (which we state but don't prove.) <br />
<br />
The difference between knowing the composition factors and knowing the group (e.g. all p-groups of the same order have the same composition factors.)<br />
<br />
'''WEEK 7'''<br />
<br />
Simplicity of A_n.<br />
<br />
Nilpotent groups (main example: the Heisenberg group)<br />
<br />
Derived series and lower central series.<br />
<br />
Category theory 101: Definition of category and functor. Some examples. A group is a groupoid with one object.<br />
<br />
'''WEEK 8'''<br />
<br />
Introduction to representation theory. <br />
<br />
<br />
<br />
Below you will find a repository of homework problems. Note that some of these problems are taken from Lang's ''Algebra''.<br />
<br />
==HOMEWORK 1 (due Sep 20)==<br />
<br />
1. Suppose that H_1 and H_2 are subgroups of a group G. Prove that the intersection of H_1 and H_2 is a subgroup of G.<br />
<br />
2. Recall that S_3 (the symmetric group) is the group of permutations of the set {1..3}. List all the subgroups of S_3.<br />
<br />
3. We can define an equivalence relation on rational numbers by declaring two rational numbers to be equal whenever they differ by an integer. We denote the set of equivalence classes by Q/Z. The operation of addition makes Q/Z into a group.<br />
<br />
a) For each n, prove that Q/Z has a subgroup of order n.<br />
<br />
b) Prove that Q/Z is a ''divisible'' group: that is, if x is an element of Q/Z and n is an integer, there exists an element y of Q/Z such that ny = x. (Note that we write the operation in this group as addition rather than multiplication, which is why we write ny for the n-fold product of y with itself rather than y^n)<br />
<br />
c) Prove that Q/Z is not finitely generated. (Hint: prove that if x_1, .. x_d is a finite subset of Q/Z, the subgroup of Q/Z generated by x_1, ... x_d is finite.)<br />
<br />
4. Let F_2 be the free group on two generators x,y. Prove that there is a automorphism of F_2 which sends x to xyx and y to xy, and prove that this automorphism is unique.<br />
<br />
5. Let H be a subgroup of G, and let N_G(H), the ''normalizer'' of H in G, be the set of elements g in G satsifying g H g^{-1} = H. Prove that N_G(H) is a subgroup of G.<br />
<br />
6. Let T be the subgroup of GL_2(R) consisting of diagonal matrices. This is an example of a "Cartan subgroup," or "torus" (whence the notation T.) Describe the normalizer N(T) of T in GL_2(R). Prove that there is a homomorphism from N(T) to Z/2Z whose kernel is precisely T. <br />
<br />
7. Prove that the group of inner automorphisms of a group is normal in the full automorphism group Aut(G).<br />
<br />
8. Let H and H' be subgroups of a group G, and let x be an element of G. We denote by HxH' the set of elements of the form hxh', with h in H and h' in H', and we call HxH a ''double coset'' of the pair (H,H').<br />
<br />
a) Show that G decomposes as a disjoint union of double cosets of (H,H').<br />
<br />
b) Let G be GL_2(R) and let B be the subgroup of upper triangular matrices. Prove that G decomposes into exactly two double cosets of (B,B).<br />
<br />
c) Let G be the symmetric group S_n, and let H be the subgroup of permutations which fix 1. Describe the double coset decomposition of S_n into double cosets of (H,H).<br />
<br />
==HOMEWORK 2 (due Sep 27)==<br />
<br />
1. Prove that the symmetric group S_n is generated by transpositions. Hint: proceed by induction on n. Let H be the subgroup of S_n consisting of permutations fixing 1. Observe that H is isomorphic to S_{n-1}, so that H is generated by transpositions. Explain why it suffices to show that, for any g in S_n, the coset gH contains a transposition. Then explain why this is the case.<br />
<br />
2. Given that the symmetric group S_n is generated by transpositions, show that there are no nontrivial homomorphisms from S_n to Z/pZ for any odd prime p.<br />
<br />
3. Compute the centralizer of the triple flip (12)(34)(56) in S_6. (Note: this is the kind of problem that a computer algebra package like sage is very good at, and I encourage you to learn to use sage well enough to check your answer.)<br />
<br />
4. Warning: a normal subgroup of a normal subgroup need not be normal! Let H be the subgroup of S_4 generated by (12)(34). Then H is a subgroup of the Klein 4-group V_4, which we have already shown is normal in S_4. Show that H is normal in V_4, but H is not normal in S_4.<br />
<br />
5. The '''quaternion group''' Q is a non-abelian finite group of order 8. Its elements are 1, -1, i, -i, j, -j, k, -k; it satisfies g^2 = -1 for all g except +-1, and ijk = 1. (Note that the notation "-i" means "the product of i with -1" and -1 is understood to be central in Q.) Prove that every subgroup of Q is normal.<br />
<br />
6. (finishing example done in class) Let Gamma be the group F<x,y> / (x^2 = 1, y^2 = 1). (Remember, this means the quotient of the free group by the group generated by all conjugates of x^2 and y^2.) Prove that Gamma is infinite. Prove that Gamma has a subgroup R generated by xy isomorphic to Z, that R is normal, and that the quotient Gamma/R is isomorphic to Z/2Z. <br />
<br />
7. Suppose a group G has the property that every element in g satisfies g^2 = 1. Show that G must be abelian.<br />
<br />
==HOMEWORK 3 (due Oct 4)==<br />
<br />
1. Multiplication by -1 is an automorphism of Z/nZ. Let a be the homomorphism Z/2Z -> Aut(Z/nZ) which sends the nontrivial element of Z/2Z to multiplication by -1. Then we have a semidirect product of Z/pZ by Z/2Z as defined in class. This group of order 2n is called a ''dihedral group.'' and is denoted D_n (or sometimes D_{2n}).<br />
<br />
1a. Compute the center of D_n. (Note that the answer depends on n!)<br />
1b. Show that all involutions of D_n are conjugate to each other if and only if n is odd.<br />
<br />
2. (More on direct products that we didn't do in class.) Suppose that G is a semidirect product of N with H. Suppose furthermore that H is also normal in G. Prove that G is a direct product of N with H.<br />
<br />
3. Let Q be the quaternion group from last week's homework. Show that Q does not decompose as a semidirect product N \rtimes H, apart form the trivial cases N = {e} and N = Q.<br />
<br />
4. The ''affine linear group'' of degree n is the group of transformations from R^n to R^n of the form x -> Ax + b, where A is a matrix in GL_n(R) and b is a vector in R^n. Prove that the affine linear group is a semidirect product R^n \rtimes GL_n.<br />
<br />
5. The ''ordinary triangle group'' T(p,q,r) is the group with presentation <x,y | x^p = y^q = (xy)^r = 1>. This is a very interesting family of groups which arises naturally in many contexts in number theory and topology.<br />
<br />
6a. Prove that T(2,2,n) is isomorphic to the dihedral group D_n.<br />
6b. Prove that T(2,3,3) is finite and compute its order. Is it isomorphic to a finite group we've discussed in class?<br />
<br />
(It is a beautiful fact, but outside the scope of this course, that T(p,q,r) is finite if and only if 1/p + 1/q + 1/r > 1.)<br />
<br />
7. Let G be a subgroup of GL_n(R) which contains SL_n(R). Prove that G is a normal subgroup of GL_n(R).<br />
<br />
8. Let G be a group and let H_1 and H_2 be subgroups of G of finite index (that is, the number of cosets in G/H_i is finite, i = 1,2.) Prove that H_1 intersect H_2 also has finite index in G.<br />
<br />
9. Recall that the commutator [x,y] is x y x^{-1} y^{-1}. If G is a group, the commutator subgroup of G, denoted G' or [G,G], is the subgroup of G generated by all commutators [x,y] for x,y in G.<br />
<br />
9a. Show that G' is a normal subgroup of G.<br />
9b. Show that G/G' is an abelian group.<br />
9c. Show that if f: G -> A is a homomorphism, with A abelian, then G' is contained in ker f. Conclude that f factors through a homomorphism G/G' -> A.<br />
9d. Show that G' = G if and only if G has no nontrivial abelian quotient. In this case, we say G is ''perfect''.<br />
<br />
10. Under what conditions on (p,q,r) is the triangle group T(p,q,r) perfect?<br />
<br />
==HOMEWORK 4 (due Oct 16)==<br />
<br />
1. Let X be a set with a transitive action of a group G, which we think of as a homomorphism f: G -> Sym(X). Let H_x be the stabilizer of an element x of X.<br />
<br />
1a. If x' is another element of X, show that H_{x"} and H_x are conjugate subgroups. (Hint: this is not true without the hypothesis of transitivity, so make sure to use it!)<br />
<br />
1b. Show that the kernel of f is the intersection of all the conjugates of H_x in G. (This subgroup is called the ''normal core'' of H_x.)<br />
<br />
2. This week, a certified mathematical grown-up asked on MathOverflow what the index-3 subgroups of SL_2(Z/pZ) were. Let's do it ourselves for homework. Let p be a prime, and suppose H is an index-3 subgroup of SL_2(Z/pZ). I do not claim that the proof sketched below is necessarily the easiest, but it allows us to cover some ground that we won't get to do in lecture.<br />
<br />
2a. Using the coset action, show that the normal core of H is the kernel of a homomorphism f from SL_2(Z/pZ) to S_3.<br />
<br />
2b. A ''unipotent matrix'' is a matrix A such that (A-I)^n = 0 for some positive integer n. Prove that a unipotent element of SL_2(Z/pZ) has order dividing p.<br />
<br />
2c. You may use the fact that unipotent matrices generate SL_2(Z/pZ). Given this fact, show that f must be trivial when p > 3.<br />
<br />
2d. Using the above, show that SL_2(Z/pZ) has no index-3 subgroups when p > 3.<br />
<br />
2e. (extra credit) What can you say about the minimal index of a proper subgroup of SL_2(Z/pZ)?<br />
<br />
3. Let H be a subgroup of G of index 2. Prove that H is normal.<br />
<br />
4. We proved two theorems in class: Lagrange's theorem, which says that the order of a subgroup of G divides |G|, and Cauchy's theorem, which says that if p divides |G| then there exists a subgroup of G of order p. Cauchy's theorem is a kind of partial converse to Lagrange's theorem, but the full converse is false. Give an example of a finite group G and a divisor n of |G| such that you can prove there is no subgroup of G of order n.<br />
<br />
5. Let X be the set of lines in F_3^2 (here a line means a 1-dimensional linear subspace) so that |X| = 4. Let G be the projective linear group PGL_2(F_3), so that G acts on X via its action on F_3^2. This action can be thought of as a homomorphism from G to S_4. Show that this homomorphism is an ISOMORPHISM from PGL_2(F_3) to S_4.<br />
<br />
6. The group S_3 has only three conjugacy classes. Prove that a finite group with at most three conjugacy classes has order at most 6.<br />
<br />
7. Let X be the set of ordered triples of elements of {1,..,n}, for some n >= 3. Then S_n acts on X. How many orbits are there? (Hint: the answer does not depend on n.)<br />
<br />
8. Suppose that G and H are groups and f: G -> H is a homomorphism. Recall the definition of the abelianization G^ab from the previous problem set.<br />
<br />
8a. Show that the composition G -> H -> H^ab factors through a unique homomorphism G^ab -> H^ab, which we denote f^ab.<br />
<br />
8b. Show that if f: G -> H and g: H -> Q are homomorphisms, then f^ab o g^ab = (f o g)^ab.<br />
<br />
(For those reading MacLane, this constitutes a proof that abelianization is a '''functor''' from the category of groups to the category of abelian groups.)<br />
<br />
==HOMEWORK 5 (due Oct 23)==<br />
<br />
1. Show that there are only two different isomorphism classes of groups of order 14. (Hint: prove that such a group must be a semidirect product of a group of order 7 by a group of order 2.)<br />
<br />
2. (variant of Burnside's Lemma) Let G be a finite group and H a normal subgroup. Suppose G acts on a finite set X. <br />
<br />
2a. Show that if x and y lie in the same H-orbit, and g is an element of G, then gx and gy also lie in the same H-orbit. Thus, G/H has a natural action on the set of H-orbits X/H.<br />
<br />
2b. Show that the average number of fixed points in X of an element of the coset gH is the same as the number of fixed points of the element gH of G/H in its action on X/H.<br />
<br />
3. A ''central extension'' of a group G by an abelian group A is a group E, together with a surjective homomorphism E -> G whose kernel is central in E and is isomorphic to A.<br />
<br />
3a. Show that the quaternion group is a central extension of (Z/2Z)^2 by Z/2Z.<br />
<br />
3b. Give an example of another nonabelian group which is a central extension of (Z/2Z)^2 by Z/2Z and which is not isomorphic to the quaternion group.<br />
<br />
4. Prove that a central extension of an abelian group is nilpotent.<br />
<br />
5. Give two different composition series for S_4 and show that they have the same composition factors.<br />
<br />
6. Suppose that A is a finitely generated abelian group, and B is a subgroup of A.<br />
Prove that rank(A/B) = rank(A) - rank(B).<br />
ADDED: I asserted in class, but did not prove, that a subgroup of a finitely generated abelian group is again finitely generated. As one class member pointed out, without this fact it's not even clear why it makes sense to talk about rank(B)! You may assume it (though it would have been better for me to have included proving this fact as part of the problem....)<br />
<br />
7. For this problem, you may use the fact (which I should have asked on homework earlier) that if H_1 and H_2 are subgroups of G, then<br />
<br />
[G: H_1 intersect H_2] <= [G:H_1][G:H_2].<br />
<br />
Prove that if G is a simple group, d is an integer greater than 1, and S is a subgroup of index d in G, then |G| <= d^d.<br />
<br />
Conclude that, in particular, A_5 has no subgroup of index 3. On the other hand, exhibit a subgroup of A_5 of index 5.<br />
<br />
(Note: d^d is not a sharp bound; can you get a bound of d factorial instead?)<br />
<br />
<!--(This argument shows that a solvable subgroup of A_n has size at most about (n-1)!, but this is far from sharp; e.g. I found a citation to a 1967 paper of Dixon which shows that a solvable subgroup of A_n has size at most k^n where k is around 2.88.)--><br />
<br />
8. Give an example of an abelian group A with a subgroup B such that B is not a direct summand of A.<br />
<br />
9. Let A be the subgroup of Z^2 generated by the vector (1,0). Let B be the subgroup of Z^2 generated by the vector (3,3). Show that A and B are isomorphic to each other, but Z^2 / A is not isomorphic to Z^2 / B.<br />
<br />
==HOMEWORK 6 (due Oct 30)==<br />
<br />
1. In this problem, you will construct a functor F from the category of finite sets to the category of complex vector spaces. I'll tell you what it does on objects: if X is a finite set, then F(X) is the vector space spanned by a set of basis elements {e_x}_{x in X}, so that dim F(X) = |X|. Your job: complete the definition of F by describing the map from Mor_{Finite Sets}(X,Y) to Mor_{Vector Spaces}(F(X),F(Y)), and showing that your map respects composition.<br />
<br />
2. If (V, rho) is a representation of a group G, denote by V^G the space of vectors in V such that gv = v for all g in G; these are called _invariant_ vectors. We saw that the permutation representation of S_3 has a one-dimensional space of invariants. Prove that if X is a finite set with G-action, and V_X the corresponding permutation representation of G, then<br />
<br />
dim V_X^G = number of orbits of X.<br />
<br />
3. (Invariants are functorial) Let G be a group. Then there is a category called C[G]-Mod whose objects are complex representations of G (i.e. pairs (V,rho) where V is a complex vector space and rho is a homomorphism from G to GL(V)). The morphisms in this category are the ones described in class: Mor((V,rho),(W,psi)) is the set of linear maps f from V to W such that<br />
<br />
f(rho(g)(v)) = psi(g)(f(v))<br />
<br />
for all g in G and all v in V.<br />
<br />
Show that there is a functor H_0 from the category of representations of G to the category of vector spaces whose action on objects is given by<br />
<br />
H_0((V) = V^G.<br />
<br />
(In other words: your job again is to explain how to associate to a morphism from (V,rho) to (W,phi) a morphism from V^G to W^G, in a way that's compatible with composition.)<br />
<br />
<!--4. The group PSL_2(Z) has a presentation of the form [U,V|U^2 = V^3 = 1]. Thus, an n-dimensional representation of PSL_2(Z) is precisely a pair of n x n matrices (rho(U),rho(V)), such that rho(U)^2 = rho(V)^3 = 1. Prove that for every n > 1 there are uncountably many isomorphism classes of n-dimensional representations of PSL_2(Z). (Extra credit: show that there are uncountably many isomorphism classes of _irreducible_ representations.)<br />
<br />
An amazing theorem of Margulis (superrigidity) shows that SL_2(Z) is very special among arithmetic groups in this way. For instance, there are only _finitely_ many isomorphism classes of n-dimensional representations of PSL_d(Z) for any d > 2; those groups are "rigid."--><br />
<br />
5. Let G be a finite group, let X be a set with G-action, and let chi be the character of the corresponding permutation representation. Show that, for each g in G, chi(g) is the number of elements of X fixed by g. (I am actually going to prove this in class.)<br />
<br />
6. Let V be the space of homogeneous degree-3 polynomials in variables x_1, x_2, x_3; then S_3 acts on V by permutation of the three variables. Describe the decomposition of V into irreducible representations of S_3.<br />
<br />
7a. Show that if chi is the character of a representation of a finite group G, then chi(g^{-1}) is the complex conjugate of chi.<br />
<br />
7b. As a consequence, show that if g is conjugate to g^{-1} for all g in G, then chi(g) is real.<br />
<br />
7c. The groups D_p and S_n have the property that g is conjugate to g^{-1} for _every_ g in G, so that every character is a real-valued function on G. Give an example of a group G where this is not the case, and give an example of a representation of G whose character takes non-real values.<br />
<br />
==HOMEWORK 7 (due Nov 6)==<br />
<br />
1. Let G be a finite group, let X be a finite set with G-action, and let chi be the character of the corresponding permutation representation. Show that, for each g in G, chi(g) is the number of elements of X fixed by g. (I am actually going to prove this in class.)<br />
<br />
2a. Show that if chi is the character of a finite-dimensional representation of a finite group G, then chi(g^{-1}) is the complex conjugate of chi(g) for all g.<br />
<br />
2b. As a consequence, show that if g is conjugate to g^{-1} for all g in G, then chi(g) is real.<br />
<br />
2c. The groups D_p and S_n have the property that g is conjugate to g^{-1} for _every_ g in G, so that every character is a real-valued function on G. Give an example of a group G where this is not the case, and give an example of a representation of G whose character takes non-real values.<br />
<br />
3. Let V and W be complex vector spaces. Let f be a linear transformation of V, and g a linear transformation of W. <br />
<br />
3a. Show that there is a unique linear transformation F satisfying<br />
<br />
F(v tensor w) = f(v) tensor g(w)<br />
<br />
for all v in V and all w in W. We denote this transformation by f tensor g.<br />
<br />
3b. Suppose V and W are finite dimensional. Show that Trace(f tensor g) = Trace(f) Trace(g). What does this say when f and g are both the identity transformation?<br />
<br />
4. Let V be a vector space. We define Sym^2 V to be the quotient of V tensor V by the subspace generated by all elements of the form<br />
<br />
v tensor w - w tensor v<br />
<br />
for v,w in V.<br />
<br />
Suppose dim V = n. What is dim Sym^2 V?<br />
<br />
5. Show that Z/pZ tensor Z/qZ is zero when p and q are distinct primes.</div>Dummithttps://hilbert.math.wisc.edu/wiki/index.php?title=741&diff=46507412012-11-02T23:48:49Z<p>Dummit: typo fixes</p>
<hr />
<div>'''Math 741'''<br />
<br />
Algebra<br />
<br />
Prof: [http://www.math.wisc.edu/~ellenber Jordan Ellenberg]<br />
<br />
Grader: Evan Dummit<br />
<br />
Ellenberg's office hours: Friday 3pm<br />
<br />
Grader's office hours: Monday 4pm [changed!]. Late homeworks may be given directly to the grader, along with either (i) the instructor's permission, or (ii) a polite request for mercy.<br />
<br />
This course, the first semester of the introductory graduate sequence in algebra, will cover the basic theory of groups, group actions, representation, linear and multilinear algebra, and the beginnings of ring theory.<br />
<br />
==SYLLABUS==<br />
<br />
In this space we will record the theorems and definitions we covered each week, which we can use as a list of notions you should be prepared to answer questions about on the Algebra qualifying exam. The material covered on the homework is also an excellent guide to the scope of the course. <br />
<br />
'''WEEK 1''': <br />
<br />
Definition of group. Associativity. Inverse. <br />
<br />
Examples of group: GL_n(R). GL_n(Z). Z/nZ. R. Z. R^*. The free group F_k on k generators. <br />
<br />
Homomorphisms. The homomorphisms from F_k to G are in bijection with G^k. Isomorphisms.<br />
<br />
'''WEEK 2''':<br />
<br />
The symmetric group (or permutation group) S_n on n letters. Cycle decomposition of a permutation. Order of a permutation. Thm: every element of a finite group has finite order. <br />
<br />
Subgroups. Left and right cosets. Lagrange's Theorem. Cyclic groups. The order of an element of a finite group is a divisor of the order of the group.<br />
<br />
The sign homomorphism S_n -> +-1.<br />
<br />
<br />
'''WEEK 3'''<br />
<br />
Normal subgroups. The quotient of a group by a normal subgroup. The first isomorphism theorem. Examples of S_n -> +-1 and S_4 -> S_3 with kernel V_4, the Klein 4-group.<br />
<br />
Centralizers and centers. Abelian groups. The center of SL_n(R) is either 1 or +-1.<br />
<br />
Groups with presentations. The infinite dihedral group <x,y | x^2 = 1, y^2 = 1>.<br />
<br />
'''WEEK 4'''<br />
<br />
More on groups with presentations.<br />
<br />
Second and third isomorphism theorems.<br />
<br />
Semidirect products.<br />
<br />
'''WEEK 5'''<br />
<br />
Group actions, orbits, and stabilizers.<br />
<br />
Orbit-stabilizer theorem.<br />
<br />
Cayley's theorem.<br />
<br />
Cauchy's theorem.<br />
<br />
'''WEEK 6'''<br />
<br />
Applications of orbit-stabilizer theorem (p-groups have nontrivial center, first Sylow theorem.)<br />
<br />
Classification of finite abelian groups and finitely generated abelian groups.<br />
<br />
Composition series and the Jordan-Holder theorem (which we state but don't prove.) <br />
<br />
The difference between knowing the composition factors and knowing the group (e.g. all p-groups of the same order have the same composition factors.)<br />
<br />
'''WEEK 7'''<br />
<br />
Simplicity of A_n.<br />
<br />
Nilpotent groups (main example: the Heisenberg group)<br />
<br />
Derived series and lower central series.<br />
<br />
Category theory 101: Definition of category and functor. Some examples. A group is a groupoid with one object.<br />
<br />
'''WEEK 8'''<br />
<br />
Introduction to representation theory. <br />
<br />
<br />
<br />
Below you will find a repository of homework problems. Note that some of these problems are taken from Lang's ''Algebra''.<br />
<br />
==HOMEWORK 1 (due Sep 20)==<br />
<br />
1. Suppose that H_1 and H_2 are subgroups of a group G. Prove that the intersection of H_1 and H_2 is a subgroup of G.<br />
<br />
2. Recall that S_3 (the symmetric group) is the group of permutations of the set {1..3}. List all the subgroups of S_3.<br />
<br />
3. We can define an equivalence relation on rational numbers by declaring two rational numbers to be equal whenever they differ by an integer. We denote the set of equivalence classes by Q/Z. The operation of addition makes Q/Z into a group.<br />
<br />
a) For each n, prove that Q/Z has a subgroup of order n.<br />
<br />
b) Prove that Q/Z is a ''divisible'' group: that is, if x is an element of Q/Z and n is an integer, there exists an element y of Q/Z such that ny = x. (Note that we write the operation in this group as addition rather than multiplication, which is why we write ny for the n-fold product of y with itself rather than y^n)<br />
<br />
c) Prove that Q/Z is not finitely generated. (Hint: prove that if x_1, .. x_d is a finite subset of Q/Z, the subgroup of Q/Z generated by x_1, ... x_d is finite.)<br />
<br />
4. Let F_2 be the free group on two generators x,y. Prove that there is a automorphism of F_2 which sends x to xyx and y to xy, and prove that this automorphism is unique.<br />
<br />
5. Let H be a subgroup of G, and let N_G(H), the ''normalizer'' of H in G, be the set of elements g in G satsifying g H g^{-1} = H. Prove that N_G(H) is a subgroup of G.<br />
<br />
6. Let T be the subgroup of GL_2(R) consisting of diagonal matrices. This is an example of a "Cartan subgroup," or "torus" (whence the notation T.) Describe the normalizer N(T) of T in GL_2(R). Prove that there is a homomorphism from N(T) to Z/2Z whose kernel is precisely T. <br />
<br />
7. Prove that the group of inner automorphisms of a group is normal in the full automorphism group Aut(G).<br />
<br />
8. Let H and H' be subgroups of a group G, and let x be an element of G. We denote by HxH' the set of elements of the form hxh', with h in H and h' in H', and we call HxH a ''double coset'' of the pair (H,H').<br />
<br />
a) Show that G decomposes as a disjoint union of double cosets of (H,H').<br />
<br />
b) Let G be GL_2(R) and let B be the subgroup of upper triangular matrices. Prove that G decomposes into exactly two double cosets of (B,B).<br />
<br />
c) Let G be the symmetric group S_n, and let H be the subgroup of permutations which fix 1. Describe the double coset decomposition of S_n into double cosets of (H,H).<br />
<br />
==HOMEWORK 2 (due Sep 27)==<br />
<br />
1. Prove that the symmetric group S_n is generated by transpositions. Hint: proceed by induction on n. Let H be the subgroup of S_n consisting of permutations fixing 1. Observe that H is isomorphic to S_{n-1}, so that H is generated by transpositions. Explain why it suffices to show that, for any g in S_n, the coset gH contains a transposition. Then explain why this is the case.<br />
<br />
2. Given that the symmetric group S_n is generated by transpositions, show that there are no nontrivial homomorphisms from S_n to Z/pZ for any odd prime p.<br />
<br />
3. Compute the centralizer of the triple flip (12)(34)(56) in S_6. (Note: this is the kind of problem that a computer algebra package like sage is very good at, and I encourage you to learn to use sage well enough to check your answer.)<br />
<br />
4. Warning: a normal subgroup of a normal subgroup need not be normal! Let H be the subgroup of S_4 generated by (12)(34). Then H is a subgroup of the Klein 4-group V_4, which we have already shown is normal in S_4. Show that H is normal in V_4, but H is not normal in S_4.<br />
<br />
5. The '''quaternion group''' Q is a non-abelian finite group of order 8. Its elements are 1, -1, i, -i, j, -j, k, -k; it satisfies g^2 = -1 for all g except +-1, and ijk = 1. (Note that the notation "-i" means "the product of i with -1" and -1 is understood to be central in Q.) Prove that every subgroup of Q is normal.<br />
<br />
6. (finishing example done in class) Let Gamma be the group F<x,y> / (x^2 = 1, y^2 = 1). (Remember, this means the quotient of the free group by the group generated by all conjugates of x^2 and y^2.) Prove that Gamma is infinite. Prove that Gamma has a subgroup R generated by xy isomorphic to Z, that R is normal, and that the quotient Gamma/R is isomorphic to Z/2Z. <br />
<br />
7. Suppose a group G has the property that every element in g satisfies g^2 = 1. Show that G must be abelian.<br />
<br />
==HOMEWORK 3 (due Oct 4)==<br />
<br />
1. Multiplication by -1 is an automorphism of Z/nZ. Let a be the homomorphism Z/2Z -> Aut(Z/nZ) which sends the nontrivial element of Z/2Z to multiplication by -1. Then we have a semidirect product of Z/pZ by Z/2Z as defined in class. This group of order 2n is called a ''dihedral group.'' and is denoted D_n (or sometimes D_{2n}).<br />
<br />
1a. Compute the center of D_n. (Note that the answer depends on n!)<br />
1b. Show that all involutions of D_n are conjugate to each other if and only if n is odd.<br />
<br />
2. (More on direct products that we didn't do in class.) Suppose that G is a semidirect product of N with H. Suppose furthermore that H is also normal in G. Prove that G is a direct product of N with H.<br />
<br />
3. Let Q be the quaternion group from last week's homework. Show that Q does not decompose as a semidirect product N \rtimes H, apart form the trivial cases N = {e} and N = Q.<br />
<br />
4. The ''affine linear group'' of degree n is the group of transformations from R^n to R^n of the form x -> Ax + b, where A is a matrix in GL_n(R) and b is a vector in R^n. Prove that the affine linear group is a semidirect product R^n \rtimes GL_n.<br />
<br />
5. The ''ordinary triangle group'' T(p,q,r) is the group with presentation <x,y | x^p = y^q = (xy)^r = 1>. This is a very interesting family of groups which arises naturally in many contexts in number theory and topology.<br />
<br />
6a. Prove that T(2,2,n) is isomorphic to the dihedral group D_n.<br />
6b. Prove that T(2,3,3) is finite and compute its order. Is it isomorphic to a finite group we've discussed in class?<br />
<br />
(It is a beautiful fact, but outside the scope of this course, that T(p,q,r) is finite if and only if 1/p + 1/q + 1/r > 1.)<br />
<br />
7. Let G be a subgroup of GL_n(R) which contains SL_n(R). Prove that G is a normal subgroup of GL_n(R).<br />
<br />
8. Let G be a group and let H_1 and H_2 be subgroups of G of finite index (that is, the number of cosets in G/H_i is finite, i = 1,2.) Prove that H_1 intersect H_2 also has finite index in G.<br />
<br />
9. Recall that the commutator [x,y] is x y x^{-1} y^{-1}. If G is a group, the commutator subgroup of G, denoted G' or [G,G], is the subgroup of G generated by all commutators [x,y] for x,y in G.<br />
<br />
9a. Show that G' is a normal subgroup of G.<br />
9b. Show that G/G' is an abelian group.<br />
9c. Show that if f: G -> A is a homomorphism, with A abelian, then G' is contained in ker f. Conclude that f factors through a homomorphism G/G' -> A.<br />
9d. Show that G' = G if and only if G has no nontrivial abelian quotient. In this case, we say G is ''perfect''.<br />
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10. Under what conditions on (p,q,r) is the triangle group T(p,q,r) perfect?<br />
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==HOMEWORK 4 (due Oct 16)==<br />
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1. Let X be a set with a transitive action of a group G, which we think of as a homomorphism f: G -> Sym(X). Let H_x be the stabilizer of an element x of X.<br />
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1a. If x' is another element of X, show that H_{x"} and H_x are conjugate subgroups. (Hint: this is not true without the hypothesis of transitivity, so make sure to use it!)<br />
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1b. Show that the kernel of f is the intersection of all the conjugates of H_x in G. (This subgroup is called the ''normal core'' of H_x.)<br />
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