> [!NOTE] Lemma ($[G:H]=2 \implies H \unlhd G$) > Let $H$ be a [[Subgroup|subgroup]] of a [[Finite Group|finite group]] $G.$ If $H$ has [[Index of Subgroup|index]] $2$ then it is [[Normal Subgroup|normal]]. ###### Proof A subgroup $H$ is normal in $G$ if and only if for all $g\in G,$ $gH=Hg.$ Suppose $g\in H.$ Then by [[Necessary Condition for Equality of Cosets|the necessary condition for equality of cosets]], $gH=H = Hg$. Suppose instead $g\not\in H.$ Then $gH \neq H$ and $Hg \neq H.$ Since [[Coset Space Partitions Subgroup|cosets partition]] $G,$ we get $gH=G\setminus H=Hg.$ # Applications **Example**: [[Alternating Group is a Normal Subgroup of Symmetric Group]].