Math 171 abstract algebra fall 2014- assignment 2 prove

Question

Math 171: Abstract Algebra, Fall 2014- Assignment 2

1 Suppose G is a group and x, g ∈ G. Prove that the order of x and g^-1xg are the same. Deduce that the order of ab is the same as the order of ba for every a, b ∈ G.

2. Let G be a group. Determine whether or not each of the following must be a subgroup of G. If true, prove it. If false, give a counterexample:

(a) The set of elements of G that commute with every element in G. This is called the center of G and is denoted Z(G). That is,

Z(G) = {h ∈ G: gh = hg for all g ∈ G}.

(b) The set of elements of G that have finite order. That is, the set

H = {h ∈ G: |h| < ∞}.

3. Determine, with justification, all subgroups of (Z, +)

4. Which of the following subsets of D₈ form a subgroup of D₈? Justify in each case.

(a) {1, r², s, sr²}.

(b) {1, r², sr, sr³}.

5. Read the "Generators and Relations" in Dummit & Foote, then prove for any n ≥ 3,

(r, s | rⁿ = s² = 1, rsr = s).

is D_2n.

6. The Heisenberg group over a field F, denoted H(F), is the set of matrices defined by

(a) Prove that H(Z/pZ) is a non-abelian subgroup of GL₃(Z/pZ) for any prime p ≥ 2.

(b) Does H(R) have a non-identity element of finite order? Prove or disprove.