How to Count Homomorphism from any finite group G to infinite cyclic group? Question How to Count Homomorphism from any finite group G to infinite cyclic group

MY Approach
I Know

I also know that a homomorphism is completely determined by its action on unit  element (like 1) 

If $x\in G$ is not the identity, there is some number $m>1$ such that $x^m$ is the identity. Since a homomorphism always maps the identity to the identity, a homomorphism $f:G'\to G$ would have $f(x^m)=f(x)^m=e$. But this means that $f(x)=e$ because the only element of finite order in an infinite cyclic group is the identity.
