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The (2,3,7) triangle group admits a presentation in terms of the group of quaternions of norm 1 in a suitable order in a quaternion algebra.More specifically, the triangle group is the quotient of the group of quaternions by its center ±1.
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 ...
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.
Geometrically, this is the symmetries of the (n − 1)-simplex, and algebraically, it yields maps and expressing these as discrete subgroups (point groups). The special orthogonal group has a 2-fold cover by the spin group Spin( n ) → SO( n ) , and restricting this cover to A n and taking the preimage yields a 2-fold cover 2⋅A n → ...
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 ...
In fact the two cyclic permutations of all three blocks, with the identity, form a subgroup of order 3, index 2, and the swaps of two blocks, each with the identity, form three subgroups of order 2, index 3. The existence of subgroups of order 2 and 3 is also a consequence of Cauchy's theorem.