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.

Recommended Page: Check out the following Recommended Math Books for Undergrads!

Author: mathtuition88

https://mathtuition88.com/

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