Subgroups of index two are normal

Statement

Let $G$ be a group and let $H$ be a subgroup of $G$ of index $2$. Then $H$ is normal in $G$.

Remarks

For $g \notin H$, the left cosets are $H$ and $gH = G \setminus H$, and the right cosets are $H$ and $Hg = G \setminus H$. Thus $gH = Hg$ for each $g \in G$, and the coset criterion applies. For example, the alternating group $A_n$ has index $2$ in the symmetric group $S_n$, and thus it is normal.

Explanations

L1 human

When a subgroup has only two cosets, one coset is the subgroup and the other is everything else. This is true for the left cosets and for the right cosets. So the left and right cosets agree, and the subgroup is normal.