741

Math 741

Algebra

Grader: Evan Dummit

Ellenberg's office hours: Friday 3pm

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.

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

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?

HOMEWORK 4 (due Oct 16)

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.

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!)

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.)

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.

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.

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.

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.

2d. Using the above, show that SL_2(Z/pZ) has no index-3 subgroups when p > 3.

2e. (extra credit) What can you say about the minimal index of a proper subgroup of SL_2(Z/pZ)?

3. Let H be a subgroup of G of index 2. Prove that H is normal.

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.

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.

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.)

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.

8a. Show that the composition G -> H -> H^ab factors through a unique homomorphism G^ab -> H^ab, which we denote f^ab.

8b. 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.)

HOMEWORK 5 (due Oct 23)

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.)

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.

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.

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.

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.

3a. Show that the quaternion group is a central extension of (Z/2Z)^2 by Z/2Z.

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.

4. Prove that a central extension of an abelian group is nilpotent.

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

6. Suppose that A is a finitely generated abelian group, and B is a subgroup of A. Prove that rank(A/B) = rank(A) - rank(B). 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....)

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

[G: H_1 intersect H_2] <= [G:H_1][G:H_2].

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.

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.

(Note: d^d is not a sharp bound; can you get a bound of d factorial instead?)

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.)

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.)

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."

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 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 representation of a finite group G, then chi(g^{-1}) is the complex conjugate of chi.

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.

741

Contents

SYLLABUS

HOMEWORK 1 (due Sep 20)

HOMEWORK 2 (due Sep 27)

HOMEWORK 3 (due Oct 4)

HOMEWORK 4 (due Oct 16)

HOMEWORK 5 (due Oct 23)

HOMEWORK 6 (due Oct 30)

HOMEWORK 7 (due Nov 6)

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741

SYLLABUS

HOMEWORK 1 (due Sep 20)

HOMEWORK 2 (due Sep 27)

HOMEWORK 3 (due Oct 4)

HOMEWORK 4 (due Oct 16)

HOMEWORK 5 (due Oct 23)

HOMEWORK 6 (due Oct 30)

HOMEWORK 7 (due Nov 6)

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