---
tag: G6FND7X69J
kind: concept
name: Coxeter group
msc: ["20F55", "51F15"]
specializes: [G5S1M9]
sources:
  - url: https://en.wikipedia.org/wiki/Coxeter_group
same_as:
  wikidata: Q7874246
  wikipedia: Coxeter group
---

## Summary

A Coxeter group is a [[group|G5S1M9]] generated by elements of order $2$.
The only relations prescribe the order of the product of each pair of generators.
The finite Coxeter groups are exactly the finite groups generated by [[reflections|06BRDF3F0E]] of a Euclidean space, such as the [[symmetric groups|0515A6D2Y1]] and the [[dihedral groups|0JZ27DMJ5D]].
Infinite Coxeter groups include the affine Weyl groups and groups generated by reflections in the sides of hyperbolic polygons whose angles have the form $\pi / m$.
They appear in Lie theory, in the theory of buildings and in geometric group theory.

## Explanations

### L1 human

Put two mirrors at an angle of $\pi / m$.
Each reflection undoes itself, so it has order $2$, and the product of the two reflections is a rotation by $2\pi / m$, of order $m$.
The group generated by the two reflections is the dihedral group of order $2m$, the symmetries of a regular $m$-gon.
A Coxeter group is any group built in the same way from several such mirrors, where only these pairwise rules are imposed.