Editing Coxeter group (section)

==Abstraction of reflection groups==
{{Further|Reflection group}}
Coxeter groups are an abstraction of reflection groups. Coxeter groups are ''abstract'' groups, in the sense of being given via a presentation. On the other hand, reflection groups are ''concrete'', in the sense that each of its elements is the composite of finitely many  geometric reflections about linear hyperplanes in some euclidean space. Technically, a reflection group is a subgroup of a [[linear group]] (or various generalizations) generated by orthogonal matrices of determinant -1. Each generator of a Coxeter group has order 2, which abstracts the geometric fact that performing a reflection twice is the identity. Each relation of the form <math>(r_ir_j)^k</math>, corresponding to the geometric fact that, given two [[hyperplane]]s meeting at an angle of <math>\pi/k</math>, the composite of the two reflections about these hyperplanes is a rotation by <math>2\pi/k</math>, which has order ''k''.

In this way, every reflection group may be presented as a Coxeter group.<ref name="Coxeter1934"/> The converse is partially true: every finite Coxeter group admits a faithful [[linear representation|representation]] as a finite reflection group of some Euclidean space.<ref name="Coxeter1935"/>
However, not every infinite Coxeter group admits a representation as a reflection group.

Finite Coxeter groups have been classified.<ref name="Coxeter1935"/>