Coxeter group

This article is about a standard (though not very rudimentary) definition in group theory. The article text may, however, contain more than just the basic definition
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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 $G$ is termed a Coxeter group if it can be equipped with a finite presentation with generators $s_{i}$ and relations:

• $s_{i}^{2}=1$
• $(s_{i}s_{j})^{m_{ij}}=1$ where $m_{ij}$ is a symmetric function of $i$ and $j$ (for distinct $i$ and $j$)

Alternatively we can consider a symmetric matrix $m_{ij}$ with the diagonal entries being $1$ and simply require that for each $i$ and $j$ (not necessarily distinct) $(s_{i}s_{j})^{m_{ij}}=1$. Note that we allow the entries $m_{ij}$ to be $0$.

Such a presentation is termed a Coxeter presentation and the matrix of $m_{ij}$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	${\begin{pmatrix}1&a\\a&1\\\end{pmatrix}}$	dihedral group of degree $a$, order $2a$.
2	${\begin{pmatrix}1&2\\2&1\\\end{pmatrix}}$	Klein four-group
2	${\begin{pmatrix}1&3\\3&1\\\end{pmatrix}}$	symmetric group of degree three
2	${\begin{pmatrix}1&4\\4&1\\\end{pmatrix}}$	dihedral group of order eight
2	${\begin{pmatrix}1&5\\5&1\\\end{pmatrix}}$	dihedral group of order ten
2	${\begin{pmatrix}1&6\\6&1\\\end{pmatrix}}$	dihedral group of order twelve
2	${\begin{pmatrix}1&8\\8&1\\\end{pmatrix}}$	dihedral group of order sixteen
3	${\begin{pmatrix}1&l&m\\l&1&n\\m&n&1\\\end{pmatrix}}$	triangle group with parameters $(l,m,n)$
3	${\begin{pmatrix}1&2&2\\2&1&2\\2&2&1\\\end{pmatrix}}$	elementary abelian group of order eight
3	${\begin{pmatrix}1&l&2\\l&1&2\\2&2&1\\\end{pmatrix}}$	Direct product of dihedral group of degree $l$ (order $2l$) and cyclic group of order two
3	${\begin{pmatrix}1&3&3\\3&1&2\\3&2&1\\\end{pmatrix}}$	symmetric group of degree four
3	${\begin{pmatrix}1&4&3\\4&1&2\\3&2&1\\\end{pmatrix}}$	direct product of S4 and Z2
3	${\begin{pmatrix}1&5&3\\5&1&3\\3&2&1\\\end{pmatrix}}$	direct product of A5 and Z2
3	${\begin{pmatrix}1&7&3\\7&1&2\\3&2&1\\\end{pmatrix}}$	(7,3,2)-triangle group	this group is infinite.
$n$	1s on diagonal, 3s on superdiagonal and subdiagonal, 2s elsewhere.	symmetric group of degree $n+1$.

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
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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.