Coxeter group

Summary

A Coxeter group is a group generated by elements of order $2$. The only relations prescribe the order of the product of each pair of generators. The finite Coxeter groups are exactly the finite groups generated by reflections of a Euclidean space, such as the symmetric groups and the dihedral groups. Infinite Coxeter groups include the affine Weyl groups and groups generated by reflections in the sides of hyperbolic polygons whose angles have the form $\pi / m$. They appear in Lie theory, in the theory of buildings and in geometric group theory.