# Definition:Commutator Subgroup

## 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**.