# Coxeter group

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This article defines a group property: a property that can be evaluated to true/false for any given group, invariant under isomorphism

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## Definition

### Symbol-free definition

A group is termed a **Coxeter group** if it can be equipped with a finite presentation given by the following relations:

- The square of every generator
- For some of the pairwise products of the generators, a certain power of that pairwise product

A presentation of this kind is termed a **Coxeter presentation**. Often, the term *Coxeter group* is used for the group along with a *specific choice* of Coxeter presentation.

### Definition with symbols

A group is termed a **Coxeter group** if it can be equipped with a finite presentation with generators and relations:

- where is a symmetric function of and (for distinct and )

Alternatively we can consider a symmetric matrix with the diagonal entries being and simply require that for each and (not necessarily distinct) . Note that we allow the entries to be .

Such a presentation is termed a **Coxeter presentation** and the matrix of s is termed a **Coxeter matrix**. Often, the term *Coxeter group* is used for a Coxeter group along with a *specific choice* of Coxeter presentation.

## Particular cases

Number of generators | Form of Coxeter matrix | Common name for the group | Comment |
---|---|---|---|

2 | dihedral group of degree , order . | ||

2 | Klein four-group | ||

2 | symmetric group of degree three | ||

2 | dihedral group of order eight | ||

2 | dihedral group of order ten | ||

2 | dihedral group of order twelve | ||

2 | dihedral group of order sixteen | ||

3 | triangle group with parameters | ||

3 | elementary abelian group of order eight | ||

3 | Direct product of dihedral group of degree (order ) and cyclic group of order two | ||

3 | symmetric group of degree four | ||

3 | direct product of S4 and Z2 | ||

3 | direct product of A5 and Z2 | ||

3 | (7,3,2)-triangle group | this group is infinite. | |

1s on diagonal, 3s on superdiagonal and subdiagonal, 2s elsewhere. | symmetric group of degree . |

## Metaproperties

### Direct products

This group property is direct product-closed, viz., the direct product of an arbitrary (possibly infinite) family of groups each having the property, also has the property

View other direct product-closed group properties

A direct product of Coxeter groups is a Coxeter group. The Coxeter matrix for the direct product is obtained by taking the block concatenation of the Coxeter matrices for the individual groups and then replacing the off-diagonal zero blocks by blocks with all entries equal to 2.