Definition:Commutator Subgroup
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Definition
Let $G$ be a group.
Its commutator subgroup $\left[{G, G}\right]$ is the subgroup generated by all commutators.
Higher derived subgroups
Let $n \ge 0$ be a natural number.
The $n$th derived subgroup $G^{\left({n}\right)}$ is recursively defined by:
- $G^{\left({n}\right)} = \begin{cases} G & : n = 0 \\ \left[{G^{\left({n - 1}\right)}, G^{\left({n - 1}\right)} }\right] & : n \ge 1 \end{cases}$
Also known as
The commutator subgroup is also known as the derived subgroup.
