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The group I is isomorphic to A 5, the alternating group on 5 letters, since its elements correspond 1-to-1 with even permutations of the five T symmetries (or the five tetrahedra just mentioned). Representing rotations with quaternions, I is made up of the 120 unit icosians. As before, this is a 1-to-2 correspondence. I h, (*532) [5,3] 5 3 2/m ...
For that group all permutations of the three non-identity elements are automorphisms, so the automorphism group is isomorphic to (which itself is isomorphic to ). In Z p {\displaystyle \mathbb {Z} _{p}} for a prime number p , {\displaystyle p,} one non-identity element can be replaced by any other, with corresponding changes in the other elements.
The unitary group is a subgroup of the general linear group GL(n, C), and it has as a subgroup the special unitary group, consisting of those unitary matrices with determinant 1. In the simple case n = 1, the group U(1) corresponds to the circle group, isomorphic to the set of all complex numbers that have absolute value 1, under multiplication ...
The isometry group of the subspace of a metric space consisting of the points of a scalene triangle is the trivial group. A similar space for an isosceles triangle is the cyclic group of order two, C 2. A similar space for an equilateral triangle is D 3, the dihedral group of order 6. The isometry group of a two-dimensional sphere is the ...
In geometry, a two-dimensional point group or rosette group is a group of geometric symmetries that keep at least one point fixed in a plane. Every such group is a subgroup of the orthogonal group O(2), including O(2) itself. Its elements are rotations and reflections, and every such group containing only rotations is a subgroup of the special ...
When the group is a continuous group, the infinitesimal generators of the group are the Killing vector fields. The Myers–Steenrod theorem states that every isometry between two connected Riemannian manifolds is smooth (differentiable). A second form of this theorem states that the isometry group of a Riemannian manifold is a Lie group.
The group Spin(3) is isomorphic to the special unitary group SU(2); it is also diffeomorphic to the unit 3-sphere S 3 and can be understood as the group of versors (quaternions with absolute value 1). The connection between quaternions and rotations, commonly exploited in computer graphics, is explained in quaternions and spatial rotations.
The complex reflection group named 3(24)3 by G.C. Shephard or 3[3]3 and by Coxeter, is isomorphic to the binary tetrahedral group. The binary tetrahedral group is most easily described concretely as a discrete subgroup of the unit quaternions, under the isomorphism Spin(3) ≅ Sp(1), where Sp(1) is the