741

Math 741

Algebra

Prof: Jordan Ellenberg

Grader: Evan Dummit

Ellenberg's office hours: Tuesday 10am

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.

The main purpose of this website is to serve as 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)

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).