User:Mosher/Draft of intro

A Coxeter group is a mathematical group defined by a presentation which says that every generator has order two, and which gives the order of the product of every pair of generators.

Coxeter groups are ubiquitous in mathematics and geometry. The finite Coxeter groups are precisely the finite Euclidean reflection groups, and they are also the finite spherical reflection groups. Consequently, the symmetry groups of all the regular polytopes are finite Coxeter groups. The Weyl groups of root systems are also all special cases of finite Coxeter groups.

Certain Coxeter groups are isomorphic to reflection groups acting on a sphere, a Euclidean or a hyperbolic space of some dimension, by a discrete group action consisting of isometries, and generated by reflections across hyperplanes. Examples include triangle groups, generated by reflections in the sides of a triangle in the two-dimensional sphere, the Euclidean plane, or the hyperbolic plane. Although not every Coxeter group is isomorphic to a reflection group of this kind, neverthless every Coxeter group is isomorphic to a linear reflection group. For these reasons, Coxeter groups are sometimes called abstract reflection groups.

Coxeter groups are named for the geometer H. S. M. Coxeter.