Surjective homomorphism onto $\mathbb Z$

Question

Let $G$ be a finitely generated group. Can we always find a surjective homomorphism $f:G\to \mathbb Z$?

I think this is true, for example if $G$ is generated by some elements that we label as $g_1,\cdots,g_n$ then set $f(g_i)=i$ for each $i=1, \dots, n$.

Answer 1

The answer is no. For example, any finite group such as the trivial group is a counterexample. If we restrict to finitely generated abelian groups, then the finite ones are the only counterexamples. For non-abelian groups, this is no longer the case. For example, the free product of any two nontrivial finite groups is an infinite non-abelian counterexample.