I'm having trouble with the next statement:
Let $G$ be a group such that there exists a surjective group homomorphism $G \rightarrow \mathbb Z$, where $\mathbb Z$ denotes the group of the integers. Then for any subgroup of finite index $H$ of $G$, there also exists a surjective group homomorphism $H\rightarrow \mathbb Z$.
I really appreciate if someone gives me a hint or a proof of this fact.