741

Math 741

Algebra

Prof: Jordan Ellenberg Grader: Evan Dummit

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.

HOMEWORK 1 (due Sep 20)

. 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.

. Recall that S_3 (the symmetric group) is the group of permutations of the set {1..3}. List all the subgroups of S_3.

. Let F_2 be the free group on two generators x,y.

a) Show that x, xyx^{-1} also generate F_2. In other words, there are plenty of pairs of elements which generate F_2; x and y are not unique generators.

b) Prove that there is an automorphism a of F_2 such that a(x) = z and a(y) = w.

. 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)