Epimorphism from G to Z

Question

I've got a problem with this exercise, I'd be thankful if someone could help. Let $G$ be a group and let $f$ be an epimorphism from $G$ to $\mathbb{Z}$. Show that for every positive integer $n$, $G$ has a normal subgroup of index $n$ in $G$. Hint: Define an epimorphism from G in $\mathbb{Z}_n$ and use the First Isomorphism Theorem.

Answer 1

More hint. Let $g$ be a composition of canonical projection from $\Bbb{Z}$ to $\Bbb{Z}/n\Bbb{Z}$ and $f$. Consider the kernel of $g$.