Definition:Derived Subgroup
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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