Coxeter group

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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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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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

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.

Relation with other properties

Stronger properties

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 simply the block concatenation of the Coxeter matrices for the individual groups.