How to get a homomorphism from $\mathbb{Z}\to G$, for $G$ a non-trivial group?

Question

Let $G$ be a group.

If $G$ is non-trivial, then there exists a non-trivial homomorphism from $\mathbb{Z}\to G$.

How would one prove the above statement?

Answer 1

Take any element $\;1\neq x\in G\;$ , and define

$$f:\Bbb Z\to G\;,\;\;f(1):=x\;\implies\;f(m):=x^m\;$$