Definition:Derived Subgroup/Higher Derived Subgroup

Definition
Let $G$ be a group.

Let $n \ge 0$ be a natural number.

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

\sqbrk {G^{\paren {n - 1} }, G^{\paren {n - 1} } } & : n \ge 1 \end {cases}$ where $\sqbrk {G^{\paren {n - 1} }, G^{\paren {n - 1} } }$ denotes the derived subgroup of $G^{\paren {n - 1} }$.

Also see

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