Group homomorphism between $\mathbb{Z}/n\Bbb Z$ and $\mathbb{Z}$ must be zero

Question

I've got to prove that any group homomorphism between $\mathbb{Z}/n\Bbb Z$ and $\mathbb{Z}$ must be zero, but I cannot seem to do it. I would appreciate any help.

Answer 1

Hint: What is the order of an element $m\in(\mathbb Z,+)$ when $m\neq 0$?

Answer 2

If $f$ is any such homomorphism, it is determined by the value of $f(1+n\mathbf Z)$). Now $$nf(1+n\mathbf Z)=f(n\cdot (1+n\mathbf Z)=f(n\mathbf Z)=0$$ since a homomorphism maps the $0$ element to the $0$ element, and thus, as $\mathbf Z$ has no zero divisors, $$f(1+n\mathbf Z)=0.$$