nLab
commutator subgroup

Definition

For $G$ a group its commutator subgroup $[G,G] \hookrightarrow G$ is the smallest subgroup containing all the group commutator elements $[g,h] \coloneqq g^{-1} h^{-1} g h$.

Properties

• The commutator subgroup is a normal subgroup. Therefore the quotient group $G^{ab} \coloneqq G/[G,G]$ exists. This is an abelian group, called the abelianization of $G$.