Math 720

Fall 2000

Homework 4

1. Let be a free group on the set .

a) Show that is a free abelian group (where is the commutator subgroup of ).

b) Use part a) to show that if is a homomorphism with an abelian group, then there is a homomorphism such that , where is the canonical projection.

c) Use parts a) and b) to give an alternative proof of the fact that every abelian group is the homomorphic image of a free abelian group.

2. Let be an abelian group.

a) Show that if is finitely generated and no nonidentity element has finite order, then is a free abelian group.

b) Is a) true if the hypothesis “finitely generated” is omitted?

c) Show that the group of positive rationals under multiplication is free abelian.

3. Prove the following theorems:

a) Every finitely generated abelian group is isomorphic to a finite direct sum of cyclic groups where the finite cyclic summands (if any) are of orders where

b) Every finitely generated abelian group is isomorphic to a direct sum of cyclic groups where each of the finite cyclic summands (if any) is of prime power order.