Question about homomorphism of cyclic groups

Question

$G$ is cyclic group. $\varphi:G\to H$ is homomorphism.

How do I show that $\ker \varphi$ and $\text{Im}\varphi$ are cyclic groups?

Thank you!

Answer 1

You can see the book for solving the second statement of your problem (see Theorem $13.9$).

By Theorem $13.3$ of the book on that link, ker$\phi$ is a normal subgroup of $G$ (In fact we only need ker$\phi$ is a subgroup of $G$) . Since $G$ is cyclic, and by the fact that subgroup of cyclic group is cyclic, we are done the first statement.

Answer 2

Hint: Note that $\ker(\phi)$ is a subgroup of $G$, so it is cyclic. Moreover, $\operatorname{im}(\phi)=\langle \phi(a)\rangle$ where $a$ is a generator of $G$.

(I have made this a community wiki answer, as it is simply Prahlad Vaidyanathan's comment. Answering this question removes it from the unanswered queue, and I dislike the other answer because it basically points at a book.)