Group Theory: Order of subgroups in Abelian groups

PROBLEM

Let $G$ be an abelian group. For any subgroup $H$ of $G$, $K$ also a subgroup of $G$, $|HK|$ divides $|G|$.

PROOF

Any subgroup of an abelian group is normal, so $H$ and $K$ are both normal subgroups of $G$ which implies $HK$ is a subgroup of $G$. The result follows from Lagrange's theorem which says that the order of any subgroup must divide the order of the group.