Editing Coxeter group (section)

===Weyl groups===
{{main|Weyl group}}
Many, but not all of these, are Weyl groups, and every Weyl group can be realized as a Coxeter group. The Weyl groups are the families <math>A_n, B_n,</math> and <math>D_n,</math> and the exceptions <math>E_6, E_7, E_8, F_4,</math> and <math>I_2(6),</math> denoted in Weyl group notation as <math>G_2.</math>

The non-Weyl ones are the exceptions <math>H_3</math> and <math>H_4,</math> and those members of the family <math>I_2(p)</math> that are not [[exceptional isomorphism|exceptionally isomorphic]] to a Weyl group (namely <math>I_2(3) \cong A_2, I_2(4) \cong B_2,</math> and <math>I_2(6) \cong G_2</math>).

This can be proven by comparing the restrictions on (undirected) [[Dynkin diagram]]s with the restrictions on Coxeter diagrams of finite groups: formally, the Coxeter graph can be obtained from the Dynkin diagram by discarding the direction of the edges, and replacing every double edge with an edge labelled 4 and every triple edge by an edge labelled 6. Also note that every finitely generated Coxeter group is an [[automatic group]].<ref name="BrinkAndHowlett">{{cite journal|last1=Brink|first1=Brigitte|last2=Howlett|first2=Robert B.|title=A finiteness property and an automatic structure for Coxeter groups|journal=Mathematische Annalen|volume=296|issue=1|pages=179–190|year=1993|doi=10.1007/BF01445101|zbl=0793.20036|s2cid=122177473}}</ref> Dynkin diagrams have the additional restriction that the only permitted edge labels are 2, 3, 4, and 6, which yields the above. Geometrically, this corresponds to the [[crystallographic restriction theorem]], and the fact that excluded polytopes do not fill space or tile the plane – for <math>H_3,</math> the dodecahedron (dually, icosahedron) does not fill space; for <math>H_4,</math> the 120-cell (dually, 600-cell) does not fill space; for <math>I_2(p)</math> a ''p''-gon does not tile the plane except for <math>p=3, 4,</math> or <math>6</math> (the triangular, square, and hexagonal tilings, respectively).

Note further that the (directed) Dynkin diagrams ''B<sub>n</sub>'' and ''C<sub>n</sub>'' give rise to the same Weyl group (hence Coxeter group), because they differ as ''directed'' graphs, but agree as ''undirected'' graphs – direction matters for root systems but not for the Weyl group; this corresponds to the [[hypercube]] and [[cross-polytope]] being different regular polytopes but having the same symmetry group.