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 known as

The derived subgroup of a group is also known as its commutator subgroup.

Also see

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

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