b. Show that a cyclic group of order 6 has a cyclic group of order 2 as a homomorphic image.

4. (a) Let G be a group such that |æ| = 2 for every x # e. Prove that G is abelian. (b) Let G be an abelian group. Prove that the set of elements of G of finite order is a subgroup of G. (c) Consider the following elements of GL2(R): a = b = -1 Show that Ja| = 3, |6| = 4, but |ab| = ∞.

On the in-class portion of Exam 1, you saw that Z5 - {[0]} is a group under multiplication. Find all the cyclic subgroups of this group.

2. (a) Let be a subgroup of the center of G. Show that if G/N is a cyclic group, then G must be abelian. (b) Show that G/Z(G) cannot be a nontrivial cyclic group.

Let m and n be integers that are greater than 1. (a) If m and n are relatively prime, prove that Zm x Z, is a cyclic group. (b) If m and n are not relatively prime, prove that Zm × Z, is not a cyclic group. (c) Construct an abelian group of order 12 that is not cyclic.