Direct Product of Cyclic Groups

Question

I have the direct product: $G_{1} \times G_{2}$ and I know it is a cyclic group. I want to prove $G_{1}$ and $G_{2}$ are themselves cyclic groups. Any help getting started?

Answer 1

Your group $G_1 \times G_2$ is cyclic, i.e. there exists $g \in G_1 \times G_2$ which generates the big group. Be aware that $g = (g_1, g_2)$ for some $g_1 \in G_1$ and $g_2 \in G_2$. So why not trying to prove your statemant with these two "obvious" candidates?

Answer 2

Any subgroup of a cyclic group is itself cyclic (do you know why?). Then you can identify $G_1$ with $G_1 \times \{1\}$ and the same with $G_2$.