User:Tomruen/Coxeter abstract groups

Coxeter abstract groups
(m, m |n, k) is an index 2 subgroup of (2, m, 2n; k), which is an index 2 subgroup of Gm,n,2q.

(l, m |n, k)
(R,S): Rl = Sm = (RS)n = (R-1S-1RS)q = 1

(l, m, n; q)
(R,S): Rl = Sm = (RS)n = (R-1S)k = 1

Gm,n,p
(A,B,C): Am = Bn = Cp = (AB)2 = (BC)2 = (CA)2 = (ABC)2 = 1

Example
Abstract groups with isomorphic Coxeter notation groups:


 * (2q, 2r| 2, p) = [[( p,q,p,r)]+]
 * (2q, 4| 2, p) = [[ p,q,p]+], with r=2
 * (6, 4| 2, 3) = [[ 3,3,3]+], 4D finite order 720
 * (6, 4| 2, 4) = [[ 4,3,4]+], 3D Euclidean
 * (6, 4| 2, 5) = [[ 5,3,5]+], 3D Hyperbolic
 * (8, 4| 2, 3) = [[ 3,4,3]+], 4D finite order 1152
 * (10, 4| 2, 3) = [[ 3,5,3]+], 3D Hyperbolic
 * (2p, 2p| 2, 2) = (4, 4| p, p) = [[ p,2,p]+], 4D finite order 4p2
 * (&infin;, &infin;| 2, 2) = (4, 4| &infin;, &infin;) = [[ &infin;,2,&infin;]+], 2D Euclidean
 * (6, 6| 2, 3) = [[( 3,3,3,3)]+], 3D Euclidean
 * (8, 6| 2, 3) = [[( 3,4,3,3)]+], 3D Hyperbolic
 * (10, 6| 2, 3) = [[( 3,5,3,3)]+], Hyperbolic
 * (8, 8| 2, 3) = (6, 6| 2, 4) = [[( 3,4,3,4)]+], Hyperbolic
 * (8, 10| 2, 3) = [[( 3,4,3,5)]+], 3D Hyperbolic
 * (10, 10| 2, 3) = (6, 6| 2, 5) = [[( 3,5,3,5)]+], 3D Hyperbolic