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Let l, m, n be integers greater than or equal to 2. A triangle group Δ(l,m,n) is a group of motions of the Euclidean plane, the two-dimensional sphere, the real projective plane, or the hyperbolic plane generated by the reflections in the sides of a triangle with angles π/l, π/m and π/n (measured in radians).
All of the discrete point symmetries are subgroups of certain continuous symmetries. They can be classified as products of orthogonal groups O(n) or special orthogonal groups SO(n). O(1) is a single orthogonal reflection, dihedral symmetry order 2, Dih 1. SO(1) is just the identity. Half turns, C 2, are needed to complete.
In fact Janko and Thompson were investigating groups similar to the Ree groups 2 G 2 (3 2n+1), and showed that if a simple group G has abelian Sylow 2-subgroups and a centralizer of an involution of the form Z/2Z×PSL 2 (q) for q a prime power at least 3, then either q is a power of 3 and G has the same order as a Ree group (it was later shown ...
A co-compact example is the (ordinary, rotational) (2,3,7) triangle group, containing the Fuchsian groups of the Klein quartic and of the Macbeath surface, as well as other Hurwitz groups. More generally, any hyperbolic von Dyck group (the index 2 subgroup of a triangle group, corresponding to orientation-preserving isometries) is a Fuchsian group.
The existence of subgroups of order 2 and 3 is also a consequence of Cauchy's theorem. The first-mentioned is { (), (RGB), (RBG) }, the alternating group A 3 . The left cosets and the right cosets of A 3 coincide (as they do for any subgroup of index 2) and consist of A 3 and the set of three swaps { (RB), (RG), (BG) }.
The three-fold axes give rise to four D 3d subgroups. The three perpendicular four-fold axes of O now give D 4h subgroups, while the six two-fold axes give six D 2h subgroups. This group is isomorphic to S 4 × Z 2 (because both O and C i are normal subgroups), and is the symmetry group of the cube and octahedron. See also the isometries of the ...
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