---
tag: 6DSWPHSWQW
kind: concept
name: Commutator theory
msc: [08A30, 08B10]
sources:
  - url: https://en.wikipedia.org/wiki/Universal_algebra
---

## Summary

Commutator theory extends the commutator of normal subgroups of a group to congruences of arbitrary algebras.
For two [[congruences|DCSXXT4SER]] $\alpha$ and $\beta$ of an algebra it defines a congruence $[\alpha, \beta]$ by a condition on term operations, the term condition.
For groups it gives the usual commutator subgroup $[M, N]$ of normal subgroups, and for rings it gives the ideal $MN + NM$.
The theory works best in [[congruence-modular varieties|QAQAS3JWXW]], where it yields notions of abelian, solvable and nilpotent algebras that behave as in group theory.

## Explanations

### L1 human

In a group, the commutator $x y x^{-1} y^{-1}$ measures how far two elements are from commuting.
Commutator theory finds the same measure in any kind of algebra, using congruences in place of normal subgroups.
An algebra whose commutator of the largest congruence with itself is trivial is called abelian; for groups these are exactly the abelian groups.
With this one notion, ideas such as solvable and nilpotent groups make sense for rings, loops and many other algebras.