Let G be a group and let H<=G with [G:H]=2 show that H normal subgroup of G
$\displaystyle H$ is a normal subgroup if and only if $\displaystyle aH=Ha$. Create the cosets $\displaystyle G/H$. If $\displaystyle a\in H$ then $\displaystyle aH=Ha$ because $\displaystyle G/H$ partitions the set $\displaystyle G$ into two sets. If $\displaystyle a\in G-H$ then $\displaystyle aH\not = H$ and $\displaystyle Ha\not = H$ thus $\displaystyle aH=Ha$ again because $\displaystyle G/H$ partitions the set into two sets.