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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 ...
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 ").
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.
Reflection. Reflections, or mirror isometries, denoted by F c,v, where c is a point in the plane and v is a unit vector in R 2.(F is for "flip".) have the effect of reflecting the point p in the line L that is perpendicular to v and that passes through c.
Isotropy group An isotropy group is the group of isomorphisms from any object to itself in a groupoid. [dubious – discuss] [1] An isotropy representation is a representation of an isotropy group. Isotropic position A probability distribution over a vector space is in isotropic position if its covariance matrix is the identity. Isotropic ...
The speed of light in vacuum is defined to be exactly 299 792 458 m/s (approximately 186,282 miles per second). The fixed value of the speed of light in SI units results from the fact that the metre is now defined in terms of the speed of light. All forms of electromagnetic radiation move at exactly this same speed in vacuum.
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 group SU(2) is isomorphic to the group of quaternions of norm 1, and is thus diffeomorphic to the 3-sphere. Since unit quaternions can be used to represent rotations in 3-dimensional space (up to sign), there is a surjective homomorphism from SU(2) to the rotation group SO(3) whose kernel is {+I, −I}.