Lemma 1   Let $(G,\cdot)$ be a group and let $H$ be a subgroup of $G$ of index 2. Then $H$ is normal in $G$

Proof. Let $G$ be a group and let $H$ be an index 2 subgroup of $G$ By definition of index, there are only two left cosets of $H$ in $G$ namely: $$H, \quad g_1H$$ where $g_1$ is any element of $G$ which is not in $H$ Notice that if $g_1,\ g_2$ are two elements in $G$ which are not in $H$ then $g_1\cdot g_2$ belongs to $H$ Indeed, the coset $g_1g_2H\neq g_1H$ (because $g_1g_2=g_1h$ would immediately yield $g_2=h\in H$ and so $g_1g_2H=H$ and $g_1g_2\in H$

Let $h\in H$ be an arbitrary element of $H$ and let $g\in G$ If $g\in H$ then $ghg^{-1}\in H$ and we are done. Otherwise, assume that $g\notin H$ Thus $gh\notin H$ and by the remark above $ghg^{-1}=(gh)g^{-1} \in H$ as desired.