A subgroup of index 2 is normal

$\textbf{Theorem}$ A subgroup of index 2 is normal. i.e.
If $H < G$ and $|G:H|=2$ then $H \lhd G$

$\textbf{Proof}$: The cosets of $G$ partition $G$ by the equivalence relation $x\sim y \Longleftrightarrow x^{-1}y \in{H}$. The equivalence class of $x$ is $[x]=xH$. Since $|G:H|=2$ there are only 2 equivalence classes, $1H$ and $aH$. But $1H=H1=H$ so the only possibility for the right coset $Ha$ is that it equals $aH$. Therefore, $H \lhd G$ $\blacksquare$

Note: a "blob" picture is useful here.