Suppose $G$ is a cyclic group and $f \colon G \rightarrow G$ is any homomorphism. Show there is an integer $n$ such that $f(\gamma) = \gamma^n$ for all $\gamma \in G$. Show that if $G$ is infinite, then $n$ is uniquely determined by $f$.
Here is my solution:
We know the image of a group homomorphism is a subgroup $H$ of $G$. So $H$ is also a cyclic group. Suppose $a$ is is a generator of $G$, then there is an integer $n$, such that $a^n$ is a generator of $H$. Because homomorphism between cyclic groups maps generator to generator (I know this result but how to prove?). Then the first statement is obvious. When $G$ is infinite, this is unique.
EDIT: Sorry for deleting this question before, because I didn't post my solution in the first version of this question.