Definition:Derived Subgroup

Definition
Let $G$ be a group.

Its commutator subgroup $\left[{G, G}\right]$ is the subgroup generated by all commutators.

Higher derived subgroups
Let $n \ge 0$ be a natural number.

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

\left[{G^{\left({n - 1}\right)}, G^{\left({n - 1}\right)} }\right] & : n \ge 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