Subgroup of Index 2 is Normal

Subgroup of Index 2 is Normal: Let $G$ be a group and let $H\le G$ be a subgroup . If the index of $H$ in $G$ is $2$, then $H$ is normal in $G$.

Equivalently: if there are exactly two left cosets of $H$ in $G$, then the left cosets equal the right cosets, so $gH=Hg$ for all $g\in G$.

Proof sketch: There are exactly two left cosets, namely $H$ and $G\setminus H$. For any $g\in G$, the right coset $Hg$ is a coset of the same size as $H$, hence must be either $H$ or $G\setminus H$. If $g\in H$ then $Hg=H=gH$. If $g\notin H$ then both $Hg$ and $gH$ are cosets distinct from $H$, hence both equal $G\setminus H$, so $Hg=gH$. Therefore $H$ is normal.