# Definition:Commutator Subgroup

## 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**.