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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 ...
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.
This group is isomorphic to A 4, the alternating group on 4 elements, and is the rotation group for a regular tetrahedron. It is a normal subgroup of T d , T h , and the octahedral symmetries. The elements of the group correspond 1-to-2 to the rotations given by the 24 unit Hurwitz quaternions (the " binary tetrahedral group ").
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 ...
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 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.
Here, the special unitary group SU(2), which is isomorphic to the group of unit norm quaternions, is also simply connected, so it is the covering group of the rotation group SO(3). Each of these covering maps are twofold covers in the sense that precisely two elements of the covering group map to each element of the quotient.
Moreover, since the unit circle is a closed subset of the complex plane, the circle group is a closed subgroup of (itself regarded as a topological group). One can say even more. The circle is a 1-dimensional real manifold , and multiplication and inversion are real-analytic maps on the circle.