## Proof that any subgroup of index 2 is normal

Let $H\leq G$ be a subgroup of index 2.

Let $g\in G$ and $h\in H$.

If $g\in H$, then $gH=H$, and $Hg=H$, hence left coset equals to right coset.

If $g\notin H$, then $gH=G\setminus H$ (set minus), and also $Hg=G\setminus H$, thus left coset also equals to right coset.

Tip: For this question, using the equivalent definition of $ghg^{-1}\in H$ to prove will be quite tricky and convoluted, as seen here.

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