741

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Math 741

Algebra

Prof: Jordan Ellenberg

Grader: Evan Dummit

Ellenberg's office hours: Tuesday 10am

Grader's office hours: Wednesday 3pm (cancelled Sep 26 and Oct 3, out of town). Late homeworks may be given directly to the grader, along with the instructor's permission, or a polite request for mercy.

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.

SYLLABUS

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.

WEEK 1:

Definition of group. Associativity. Inverse.

Examples of group: GL_n(R). GL_n(Z). Z/nZ. R. Z. R^*. The free group F_k on k generators.

Homomorphisms. The homomorphisms from F_k to G are in bijection with G^k. Isomorphisms.

WEEK 2:

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.

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.

The sign homomorphism S_n -> +-1.


WEEK 3

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.

Centralizers and centers. Abelian groups. The center of SL_n(R) is either 1 or +-1.

Groups with presentations. The infinite dihedral group <x,y | x^2 = 1, y^2 = 1>.

WEEK 4

More on groups with presentations.

Second and third isomorphism theorems.

Semidirect products.





Below you will find a repository of homework problems. Note that some of these problems are taken from Lang's Algebra.

HOMEWORK 1 (due Sep 20)

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.

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.

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.

a) For each n, prove that Q/Z has a subgroup of order n.

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)

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.)

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.

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.

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.

7. Prove that the group of inner automorphisms of a group is normal in the full automorphism group Aut(G).

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').

a) Show that G decomposes as a disjoint union of double cosets of (H,H').

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).

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).

HOMEWORK 2 (due Sep 27)

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.

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.

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.)

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.

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.

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.

7. Suppose a group G has the property that every element in g satisfies g^2 = 1. Show that G must be abelian.

HOMEWORK 3 (due Oct 4)

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}).

1a. Compute the center of D_n. (Note that the answer depends on n!) 1b. Show that all involutions of D_n are conjugate to each other if and only if n is odd.

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.

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.

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.

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.

6a. Prove that T(2,2,n) is isomorphic to the dihedral group D_n. 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?

(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.)

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).

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.

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.

9a. Show that G' is a normal subgroup of G. 9b. Show that G/G' is an abelian group. 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. 9d. Show that G' = G if and only if G has no nontrivial abelian quotient. In this case, we say G is perfect.

10. Under what conditions on (p,q,r) is the triangle group T(p,q,r) perfect?