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 $[g,h] \coloneqq g^{-1} h^{-1} g h$.

This definition also makes sense for invertible semigroups, for which they would be called ‘commutator invertible subsemigroups’.

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 $G$.

Related concepts

• abelianization