commutator subgroup

- group, ∞-group
- group object, group object in an (∞,1)-category
- abelian group, spectrum
- group action, ∞-action
- representation, ∞-representation
- progroup
- homogeneous space

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’.

- 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 $G$.

Last revised on June 14, 2021 at 11:13:20. See the history of this page for a list of all contributions to it.