Definition:Commutator Subgroup

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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

Sources

• 1989: Ephraim J. Borowski and Jonathan M. Borwein: Dictionary of Mathematics ... (previous) ... (next): derived subgroup