Derived subgroup of abelian group is trivial

Statement

The derived subgroup of an abelian group is the trivial group.

Proof

If $G$ is abelian, then for any $g,h\in G,g^{-1}h^{-1}gh=g^{-1}gh^{-1}h=ee=e$. Thus the derived subgroup $G'=\{g^{-1}h^{-1}gh:g,h\in G\}$ is trivial.

Converse

The converse derived subgroup is trivial implies group is abelian holds as well. Thus the derived subgroup is trivial if and only if group is abelian.