( ) If contains all the commutators of , then and is abelian.

Assume K contains all the commutators of . Then for all . It follows that . Thus, for all . Therefore, is abelian.

We remain to show that K is normal. To show that K is normal, we need to show that for all , . Let for . Then, we need to check is in K. , which is in K. Thus, .

( ) If and is abelian, then contains all the commutators of .

Assume and is abelian. Then, for all . It follows that . Thus, K contains all the commutator of G.