Homework 2
Math 332, Spring 2013
These problems must be written up in LATEX, and are due this Friday, February 15.
1. Let G = (0, ∞) × R, and define a binary operation ∗ on G by the formula
(a, b) ∗ (c, d) = (ac, bc + d)
for all (a, b), (c, d) ∈ G. Prove that G forms a group under the operation ∗.
2. Prove that every finite group with an even number of elements has at least one element
of order two.
3. Let G be a group. Given elements a, b ∈ G, we say that b is conjugate to a if there
exists an element c ∈ G such that b = c−1 ac.
(a) Prove that conjugacy is an equivalence relation on G.
(b) Determine the equivalence classes (known as conjugacy classes) when G is the
group of symmetries of an equilateral triangle.
(c) Determine the conjugacy classes when G is the group of symmetries of a square.
4. Let G be a group, and suppose that a2 = e for every element a ∈ G. Prove that G is
abelian.