$\displaystyle \\\mbox{Let $G$ be a finite group and $H$ a subgroup}$

$\displaystyle \\\mbox{such that $|H|$ is odd and }|G:H|=2.$

$\displaystyle \\\mbox{Show that }\forall\,a,b\in G,\ ab\in H\ \Leftrightarrow\ \mbox{the orders}$

$\displaystyle \\\mbox{of $a$ and $b$ are both odd or both even}.$