---
tag: 0H9K5D3RDV
kind: definition
name: Abelianisation
msc: [20F12]
defines: 0AQZV0905N
uses: [08SJ5K2CBG, 0YYWCNTPWR]
sources:
  - url: https://en.wikipedia.org/wiki/Commutator_subgroup
    locator: Abelianization
---

## Definition

Let $G$ be a [[group|G5S1M9]] and let $[G, G]$ be its [[commutator subgroup|0ASGSTXSBW]].
The abelianisation of $G$ is the [[quotient group|0KYX2N69J0]] $G^{\mathrm{ab}} = G/[G, G]$.

## Remarks

The group $G^{\mathrm{ab}}$ is [[abelian|082GHNPJ53]], by [[this criterion|0MS8QGA6S5]].
If $G$ is abelian, then $G^{\mathrm{ab}} = G$.
The abelianisation of the symmetric group $S_n$ for $n \geq 2$ is cyclic of order $2$, and the sign of a permutation gives the isomorphism.

## Explanations

### L1 human

The abelianisation forces all elements of a group to commute, and changes nothing more than it must.
It keeps as much information about the group as an abelian group can keep.
For a group that is already abelian, it does nothing.