# Changes

The maximal abelian quotient of any group is termed its [[abelianization]], and this is the quotient by the [[commutator subgroup]]. A subgroup is an [[abelian-quotient subgroup]] (i.e., normal with abelian quotient group ) if and only if the subgroup contains the commutator subgroup. ==Formalisms== {{obtainedbyapplyingthe|diagonal-in-square operator|normal subgroup}} A group $G$ is an abelian group if and only if, in the [[external direct product]] $G \times G$, the diagonal subgroup $\{ (g,g) \mid g \in G \}$ is a [[normal subgroup]].