Group Theory - Cyclic Groups & Direct Product

Question

Prove that if $G$ is an infinite group and $H$ is a group then $G \times H$ is cyclic if and only if $G$ is cyclic a $H =$ {${e_H}$}.

Solution: I can see that this is true but I don't know how to start my proof.

Answer 1

Since $G\times H$ is cyclic, let $\langle(g,h)\rangle=G\times H$, so $G=\langle g\rangle$ and $H=\langle h\rangle$ are cyclic as well. Let $H=\mathbb{Z}_n$ for some positive integer $n$. Then, if $h^2\neq h$, $(g^2,h)\not\in\langle(g,h)\rangle$; otherwise, $G$ is finite. Therefore, $h^2=h$, or $h=e_H$.