If $G$ is an infinite group such that $G$ is a homomorphic image of $\Bbb Z$ then prove that $G$ is isomorphic to $\Bbb Z$

Question

If $G$ is an infinite group such that $G$ is a homomorphic image of $\Bbb Z$ then prove that $G$ is isomorphic to $\Bbb Z$.

In this question, I tried applying the First Isomorphism Theorem. But I don’t know what the injective homomorphism will be. If we consider $G=\{g_1,g_2,g_3,…\}$, then can we consider $f(g_k)=k$, $k\in\Bbb Z$, a homomorphism? If not, then why?

I don’t know how to approach this problem.

I think there might be posts concerning the same topic, but I couldn’t find any. But still, I want to know $f$ as described above, can be considered a homomorphism or not?

Answer 1

Any homomorphic image of a cyclic group is cyclic. So $G$ is infinite cyclic. Thus $G\cong\Bbb Z.$