Definition:Derived Subgroup

Definition

Let $G$ be a group.

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

Higher Derived Subgroup

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}$

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

• Results about derived subgroups can be found here.

Sources

• 1989: Ephraim J. Borowski and Jonathan M. Borwein: Dictionary of Mathematics ... (previous) ... (next): commutator: 1.
• 1989: Ephraim J. Borowski and Jonathan M. Borwein: Dictionary of Mathematics ... (previous) ... (next): derived subgroup
• 2021: Richard Earl and James Nicholson: The Concise Oxford Dictionary of Mathematics (6th ed.) ... (previous) ... (next): commutator