Commutator subgroup is the unique smallest normal subgroup such that is abelian.
If is a group, then is a normal subgroup of and is abelian. If is a normal subgroup of , then is abelian iff contains .
Let be any automorphism. Then
It follows that . In particular, if is the automorphism given by conjugation by , then , so .
Since , and hence is abelian.
() If is abelian, then for all . Hence . Therefore, contains all commutators and .
() If , then . Thus for all . Hence is abelian.