If $A$ and $B$ are finite subgroups, of orders $m$ and $n$, respectively, of the abelian group $G$, prove that $AB$ is a subgroup of order $mn$ if $m$ and $n$ are relatively prime.
Lagrange's theorem has not been introduced in this part of the book, so please refrain from using it. I think I managed to prove it by considering the minimal generating set of the subgroups. But my proof is quite long, while this is supposedly middle-level question of the problem set. So I expect to see some simple proofs.