Coxeter group

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:

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) $$(s_is_j)^{m_{ij}} = 1$$. Note that we allow the entries $$m_{ij}$$ to be $$0$$.
 * $$s_i^2 = 1$$
 * $$(s_is_j)^{m_{ij}} = 1$$ where $$m_{ij}$$ is a symmetric function of $$i$$ and $$j$$ (for distinct $$i$$ and $$j$$)

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.

Metaproperties
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.