Let H be a group and K a subgroup of H with (H:K)=n< ∞.
Prove: if (H:K)=2, then K is a normal subgroup of H
The subgroup K partitions H into left cosets, one of which is K itself. If (H:K)=2, then there is only one other left coset, namely everything in H that isn't in K. The same applies to right cosets. Thus when (H:K)=2, each left coset is also a right coset. But that is one way of expressing the condition for K to be normal.