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Such subgroups are sometimes referred to as "ordinary" triangle groups [2] or von Dyck groups, after Walther von Dyck. For spherical, Euclidean, and hyperbolic triangles, these correspond to the elements of the group that preserve the orientation of the triangle – the group of rotations. For projective (elliptic) triangles, they cannot be so ...
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 (2,3,7) triangle group is defined as the index 2 subgroups consisting of the orientation-preserving isometries, which is a Fuchsian group (orientation-preserving NEC group). Uniform heptagonal/triangular tilings
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.
More generally, any hyperbolic von Dyck group (the index 2 subgroup of a triangle group, corresponding to orientation-preserving isometries) is a Fuchsian group. All these are Fuchsian groups of the first kind. All hyperbolic and parabolic cyclic subgroups of PSL(2,R) are Fuchsian. Any elliptic cyclic subgroup is Fuchsian if and only if it is ...
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 ...
It fixes a 2-2-3 triangle in the Leech lattice and thus is a subgroup of the Conway groups, , and . Its Schur multiplier has order 3, and its outer automorphism group has order 2. The group 3.McL:2 is a maximal subgroup of the Lyons group.
The intersection of subgroups A and B of G is again a subgroup of G. [5] For example, the intersection of the x-axis and y-axis in under addition is the trivial subgroup. More generally, the intersection of an arbitrary collection of subgroups of G is a subgroup of G.