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If $ H $ is a subgroup of index 2, then it has two cosets in $ G $, which we may denote by $ H $ and $ G – H $. If $ g \in H $, then clearly we have $ gH = Hg = H $. Otherwise, $ gH, Hg \neq H $ since $ g \notin H $, however any coset of $ H $ is either $ H $ or $ G – H $, so we must have that $ gH = Hg = G – H $. Since the left and right cosets coincide, we conclude that $ H $ is normal.