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 $[g,h]≔g^{-1}{h}^{-1}gh$.

Properties

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