---
tag: 08SJ5K2CBG
kind: definition
name: Commutator subgroup
msc: [20F12]
defines: 0ASGSTXSBW
uses: [06RGT4V6Y6, 0RG9SPQSWM]
sources:
  - url: https://en.wikipedia.org/wiki/Commutator_subgroup
    locator: Definition
---

## Definition

Let $G$ be a [[group|G5S1M9]].
The commutator subgroup of $G$ is the [[subgroup generated by|0RP7VWQ0YN]] the set of all [[commutators|0PXW0BN7DW]] $[g, h]$ with $g, h \in G$.
It is written $[G, G]$ or $G'$.

## Remarks

A product of commutators need not be a commutator, so in general the set of commutators is not a subgroup.
The subgroup $[G, G]$ is [[characteristic|0KABPMRHM2]], so it is normal in $G$.
The group $G$ is abelian if and only if $[G, G] = \{e\}$.
The commutator subgroup of the symmetric group $S_n$ is the alternating group $A_n$.

## Explanations

### L1 human

The commutator subgroup collects all the failures of commutativity in a group.
If you divide the group by it, you get an abelian group.
A large commutator subgroup means a group that is far from abelian.