---
tag: 06RGT4V6Y6
kind: definition
name: Commutator of two elements
aliases: [group-commutator]
msc: [20F12]
defines: 0PXW0BN7DW
uses: [G5S1M9]
sources:
  - url: https://en.wikipedia.org/wiki/Commutator
    locator: Group theory
---

## Definition

Let $G$ be a [[group|G5S1M9]] and let $g, h \in G$.
The commutator of $g$ and $h$ is the element $[g, h] = g^{-1} h^{-1} g h$ of $G$.

## Remarks

The equation $gh = hg [g, h]$ holds, so $[g, h] = e$ if and only if $gh = hg$.
Some authors use the convention $[g, h] = g h g^{-1} h^{-1}$ instead.
The inverse of $[g, h]$ is $[h, g]$.

## Explanations

### L1 human

The commutator measures how far two elements are from commuting.
It is the correction factor that you need when you swap $g$ and $h$ in a product.
In an abelian group, every commutator is the identity.