Index-two subgroup is normal
= Index-two subgroup is normal
Every subgroup $H\leq G$ of index two is normal: the two left cosets and the two right cosets are both $H$ and its complement, so they agree.
= Index-two subgroup is normal
Every subgroup $H\leq G$ of index two is normal: the two left cosets and the two right cosets are both $H$ and its complement, so they agree.