In mathematics, a Coxeter group, named after H. S. M. Coxeter, is an abstract group that admits a formal description in terms of reflections (or kaleidoscopic mirrors). Indeed, the finite Coxeter groups are precisely the finite Euclidean reflection groups; for example, the symmetry group of each regular polyhedron is a finite Coxeter group. However, not all Coxeter groups are finite, and not all can be described in terms of symmetries and Euclidean reflections. Coxeter groups were introduced in 1934 as abstractions of reflection groups, and finite Coxeter groups were classified in 1935.

Coxeter groups find applications in many areas of mathematics. Examples of finite Coxeter groups include the symmetry groups of regular polytopes, and the Weyl groups of simple Lie algebras. Examples of infinite Coxeter groups include the triangle groups corresponding to regular tessellations of the Euclidean plane and the hyperbolic plane, and the Weyl groups of infinite-dimensional Kac–Moody algebras.

Formally, a Coxeter group can be defined as a group with the presentation

where ${\displaystyle m_{ii}=1}$ and ${\displaystyle m_{ij}=m_{ji}\geq 2}$ is either an integer or ${\displaystyle \infty }$ for ${\displaystyle i\neq j}$. Here, the condition ${\displaystyle m_{ij}=\infty }$ means that no relation of the form ${\displaystyle (r_{i}r_{j})^{m}=1}$ for any integer ${\displaystyle m\geq 2}$ should be imposed.

The pair ${\displaystyle (W,S)}$ where ${\displaystyle W}$ is a Coxeter group with generators ${\displaystyle S=\{r_{1},\dots ,r_{n}\}}$ is called a Coxeter system. Note that in general ${\displaystyle S}$ is not uniquely determined by ${\displaystyle W}$. For example, the Coxeter groups of type ${\displaystyle B_{3}}$ and ${\displaystyle A_{1}\times A_{3}}$ are isomorphic but the Coxeter systems are not equivalent, since the former has 3 generators and the latter has 1 + 3 = 4 generators (see below for an explanation of this notation).

A number of conclusions can be drawn immediately from the above definition.

The reason that ${\displaystyle m_{ij}=m_{ji}}$ for ${\displaystyle i\neq j}$ is stipulated in the definition is that

together with