Definition:Derived Subgroup

Definition
Let $G$ be a group.

Its commutator subgroup $[G, G]$ is the subgroup generated by all commutators.

Higher derived subgroups
Let $n\geq0$ be a natural number.

The $n$th derived subgroup $G^{(n)}$ is recursively defined by:
 * $G^{(n)} = \begin{cases}G & : n = 0 \\

\left[ G^{(n-1)}, G^{(n-1)} \right] & : n \geq 1\end{cases}$

Also known as
The commutator subgroup is also known as the derived subgroup.

Also see

 * Commutator Subgroup is Characteristic Subgroup
 * Definition:Abelianization of Group
 * Definition:Derived Series of Group