Finding a subgroup of an abelian group that is isomorphic to Z

Question

The question: If G is an abelian group and f is a surjective homomorphism from G to Z with kernel K, prove that G has a subgroup H such that H is isomorphic to Z.

By the first isomorphism theorem I know that G/K is isomorphic to Z (as Z is the image of f). So I figure I just have to find an H that is isomorphic to G/K, and I'm good (thanks to the transitivity of isomorphisms).

Answer 1

Hint: You only need to prove that $G$ contains an element of infinite order.

Solution:

If $\phi:G \to \mathbb Z$ is a surjective homomorphism, take $g \in G$ such that $\phi(g)=1$.