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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''.
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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.
 
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Revision as of 20:44, 2 September 2013

Math 741

Algebra

Prof: Jordan Ellenberg

Grader: Evan Dummit

Ellenberg's office hours: Friday 3pm.

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.

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.

WEEK 5

Group actions, orbits, and stabilizers.

Orbit-stabilizer theorem.

Cayley's theorem.

Cauchy's theorem.

WEEK 6

Applications of orbit-stabilizer theorem (p-groups have nontrivial center, first Sylow theorem.)

Classification of finite abelian groups and finitely generated abelian groups.

Composition series and the Jordan-Holder theorem (which we state but don't prove.)

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.)

WEEK 7

Simplicity of A_n.

Nilpotent groups (main example: the Heisenberg group)

Derived series and lower central series.

Category theory 101: Definition of category and functor. Some examples. A group is a groupoid with one object.

WEEK 8

Introduction to representation theory.

WEEK 10

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.

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.


8. Give an example of an abelian group A with a subgroup B such that B is not a direct summand of A.

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.

HOMEWORK 6 (due Oct 30)

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.

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

dim V_X^G = number of orbits of X.

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

f(rho(g)(v)) = psi(g)(f(v))

for all g in G and all v in V.

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

H_0((V) = V^G.

(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.)


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.)

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.

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.

7b. As a consequence, show that if g is conjugate to g^{-1} for all g in G, then chi(g) is real.

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.

HOMEWORK 7 (due Nov 6)

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.)

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.

2b. As a consequence, show that if g is conjugate to g^{-1} for all g in G, then chi(g) is real.

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.

3. Let V and W be complex vector spaces. Let f be a linear transformation of V, and g a linear transformation of W.

3a. Show that there is a unique linear transformation F satisfying

F(v tensor w) = f(v) tensor g(w)

for all v in V and all w in W. We denote this transformation by f tensor g.

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?

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

v tensor w - w tensor v

for v,w in V.

Suppose dim V = n. What is dim Sym^2 V?

5. Show that Z/pZ tensor Z/qZ is zero when p and q are distinct primes.

HOMEWORK 8 (due Nov 13)

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.

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.)

1b. Show that the map from V to V induced by g is G-equivariant, but is neither zero nor a scalar multiplication.

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.

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

<chi_{V_1}, chi_{V_2}> = dim_C Hom_G(V_1,V_2)

where we recall that Hom_G(V_1,G_2) refers to the space of G-equivariant homomorphisms from V_1 to V_2.

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!)

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.

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.)

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.)

5c. Using the decomposition of the regular representation, show that the irreducible representations of G you have constructed are the only ones.

HOMEWORK 9 (due Nov 20)

0. Find all ring homomorphisms from Z to Z/30Z. [Warning: there are 8 of them, not 30.]


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.

1a. Show that any nonzero Boolean ring has characteristic 2. (In other words, show that a+a=0 for all a in R.)

1b. Show that every Boolean ring is commutative.


2. Let phi : R -> S be a homomorphism of commutative rings with 1.

2a. If I is an ideal of S, show that phi^(-1) (I) is an ideal of R.

2b. If P is a prime ideal of S, show that phi^(-1) (P) is either R or a prime ideal of R.

2c. If M is a maximal ideal of S and phi is surjective, show that phi^(-1) (M) is a maximal ideal of R.

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.


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.

3a. Find (a generator of) Nil(Z/720Z).

3b. Show that Nil(R/Nil(R)) = 0.

3c. If x is in Nil(R) and u is a unit, show that u+x is a unit.

3d. Show that Nil(S), where S is the 2x2 matrices with real coefficients, is not an ideal of S.


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.

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.


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.


6. Prove that the direct sum of any collection of free R-modules is also a free R-module.


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.

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.]

7b. Show that M/N1 is a free Z-module.

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.]

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.

7e. Use problem 5 to obtain a contradiction.


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,...).

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.]

8b. Show that R is isomorphic to R^n for every positive integer n.


HOMEWORK 10 (due Dec 4)

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

f(x+e) - f(x) = eg(x)

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!

2. Write down all the ideals of C[e]/e^2. (There are three.)

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.


4. An idempotent element of a ring is an element e satisfying e^2 = e.

4a. If e is an idempotent, show that 1-e is also an idempotent.

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.)

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

f_{e} x f_{1-e}: R -> eR x (1-e)R

is actually an isomorphism of rings. In other words, the presence of a central idempotent induces a decomposition a ring as a direct product.


5. The center of a ring R is the subring of elements commuting with every element of R.

5a. Show that the center of M_n(Q) is the ring of scalar matrices.

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.


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.

6a. Describe the submodule of V which is annihilated by the element d-x in W.

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!)

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.)


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.


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.

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).

8c. Let V,W be as in question 6. Show that Ann(e^x) is the left ideal W(d-1).

HOMEWORK 11 (due Dec 13)

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.

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.

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).

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.

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."

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.)

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.

3c. (Optional for people who like propositional logic) Show that a(b+c) = ab + ac with the given operations.


4. Let K be Q[x]/(x^3-2).

4a. Show that K is a field.

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!]

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.

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.

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.

5c. Construct a ring isomorphism between M_n(Q) and M_n(Q)^op.

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!)

6. . Let R be a commutative ring with 1. If R has a unique maximal ideal M, R is called a "local ring".

6a. If R is a local ring, show that every element of R that is not in M is a unit.

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.

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.

7a. Show that dim_C (V tensor_C[S_3] C[S_4]) = 4 dim_C V.

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.

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