I want to show $H$ is a subgroup of $G$, an Abelian group, where $H=\{x \in G | |x| \ is \ odd \}$. If I assume $a,b\in H$ can I be certain $ab^{-1} \in H$, that is to say does multiplying two elements with odd order in an Abelian group produce an element with odd order? I've tried it out with a few specific groups and it seems to be the case but I haven't been able to come up with a proof.