Definition:Derived Subgroup

Definition
Let $G$ be a group.

Its commutator subgroup $\sqbrk {G, G}$ is the subgroup generated by all commutators.

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

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

\sqbrk {G^{\paren {n - 1} }, G^{\paren {n - 1} } } & : 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