# 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):**derived subgroup**