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): Entry: derived subgroup
