TITLE: Are all normal subgroups Abelian?
QUESTION [12 upvotes]: If $H \subset G$ is a normal subgroup of G,
=> $xHx^{-1} = H$ or $xH = Hx$ for all $x \epsilon G$
=> $xH = Hx$ for all $x \epsilon H$
Hence, all normal subgroups of a group are themselves Abelian?
Also, does that mean that all normal towers of subgroups are also Abelian?

REPLY [2 votes]: To get other counterexamples, $A$ and $B$ are normal subgroups in the direct product $A \times B$.