# Difference between revisions of "741"

Line 342: | Line 342: | ||

4b. Let R be the ring of continuous functions from the real numbers to the real numbers. Show that R is not Noetherian. | 4b. Let R be the ring of continuous functions from the real numbers to the real numbers. Show that R is not Noetherian. | ||

− | 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 | + | 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 |

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

Line 351: | Line 351: | ||

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. | 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. | ||

− | |||

− | |||

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

## Revision as of 10:51, 1 November 2013

**Math 741**

Algebra

Prof: Jordan Ellenberg

Grader: Evan Dummit.

* Homework policies: * Late homework may be given directly to the grader, along with either (i) the instructor's permission, or (ii) a polite request for mercy. * Assignments that are more than 1 page should be affixed in some reasonable way. * Results from places (e.g., the internet) other than 741 and standard books must be cited.

JE's office hours: Monday 12pm-1pm (right after class)

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.

## HOMEWORK 1 (due Sep 16)

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.)

d) Conclude that Q is not finitely generated.

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.

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).

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).

c) Explain why a) and b) imply that f must be identically 0.

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.

## HOMEWORK 2 (due Sep 25)

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.

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.

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.

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.

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

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').

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). (This is an example of the so-called *Bruhat decomposition* which is of great importance in the theory of algebraic groups and their representations.)

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 3 (due Oct 2)

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.)

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.

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.

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.

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.

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.

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?

## HOMEWORK 4 (due Oct 9)

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.

1a. Show that G' is a normal subgroup of G.
1b. Show that G/G' is an abelian group, which we call the abelianization, G^ab.
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.
1d. Show that G' = G if and only if G has no nontrivial abelian quotient. In this case, we say G is *perfect*.

2. Prove that any p-Sylow subgroup of GL_n(F_p) is contained in SL_n(F_p).

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).

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)?

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).

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.

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.

7a. Show that the composition G -> H -> H^ab factors through a unique homomorphism G^ab -> H^ab, which we denote f^ab. 7b. Show that if f: G -> H and g: H -> Q are homomorphisms, then f^ab o g^ab = (f o g)^ab.

(For those reading MacLane, this constitutes a proof that abelianization is a **functor** from the category of groups to the category of abelian groups.)

8. Give two different composition series for S_4 and show that they have the same composition factors.

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.

## HOMEWORK 5 (due Oct 16)

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.

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.)

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.)

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.

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.

6b. As a consequence, show that if g is conjugate to g^{-1} for all g in G, then chi(g) is real.

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.

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.)

## HOMEWORK 6 (due Oct 23)

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.

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.

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.)

2b. Show that the map from V to V induced by g is G-equivariant, but is neither zero nor a scalar multiplication.

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.

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. 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.

6. Give an example showing that the standard representation of S_n need NOT be irreducible over a field of characteristic p.

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}.

## HOMEWORK 7 (due Oct 30)

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

<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.

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).

We will work out all the irreducible representations of G.

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.)

2b. Now show that there are 4 1-dimensional representations of G.

2c. Show that the irreducible representations of G you have constructed are the only ones.

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

chi_{V tensor W}(g,h) = chi_V(g) chi_W(h).

3a. Prove that V tensor W is irreducible if and only if V and W are irreducible.

3b. Prove that all the irreducible representations of G x H are of the form V tensor W.

(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.)

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.

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.

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.

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.)

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

gf(x) = f(x+a)

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!)

5b. There is a natural norm on V which sends f to ||f|| = sum_x |f(x)|^2.

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.

Prove that ||f|| = sum_i ||f_i||.

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.

## HOMEWORK 8 (due Nov 6)

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 (recalling that rings for us have a multiplicative identity 1 and that ring homomorphisms take 1 to 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(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.

3a. Show that Nil(R) is an ideal whenever R is a commutative ring.

3b. Describe Nil(Z/720Z).

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

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.)

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."

4a. Show that Z is Noetherian.

4b. Let R be the ring of continuous functions from the real numbers to the real numbers. Show that R is not Noetherian.

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

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!

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

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.