 Normal subgroup of order two is central

If $N=\{1,z\}$ is a normal subgroup of a group $G$, conjugation by any element of $G$ preserves $N$ and fixes $1$. It must therefore fix $z$ as well. Thus $N$ lies in the center of a group. This fact can turn a small normal subgroup into a commuting factor without assuming the whole group is abelian.