# 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 .

## Proof

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.