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.


Also see