Question:Subgroup of direct product direct product of subgroups

This question is about external direct product, subgroup| See more questions about external direct product

This question has type complete description conjecture

Q: Consider the external direct product $G_{1}\times G_{2}$. Is every subgroup of this of the form $H_{1}\times H_{2}$ where $H_{1}$ is a subgroup of $G_{1}$ and $H_{2}$ is a subgroup of $G_{2}$, embedded in the obvious way inside $G_{1}\times G_{2}$?

A: Not in general. In particular, if we take $G_{1}=G_{2}=G$ and consider the group $G\times G$, this has a diagonal subgroup:

$\{(g,g)\mid g\in G\}$

which is not of the indicated form.

However, the statement is true in some cases. For finite groups, it is true in the case that $G_{1}$ and $G_{2}$ have relatively prime orders to each other. (reference needed).