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The rotation group is a Lie group of rotations about a fixed point. This (common) fixed point or center is called the center of rotation and is usually identified with the origin. The rotation group is a point stabilizer in a broader group of (orientation-preserving) motions. For a particular rotation: The axis of rotation is a line of its ...
The rotation group is a group under function composition (or equivalently the product of linear transformations). It is a subgroup of the general linear group consisting of all invertible linear transformations of the real 3-space. [2] Furthermore, the rotation group is nonabelian. That is, the order in which rotations are composed makes a ...
In geometry the rotation group is the group of all rotations about the origin of three-dimensional Euclidean space R 3 under the operation of composition. [1] By definition, a rotation about the origin is a linear transformation that preserves length of vectors (it is an isometry) and preserves orientation (i.e. handedness) of space.
This group is also called the rotation group, generalizing the fact that in dimensions 2 and 3, its elements are the usual rotations around a point (in dimension 2) or a line (in dimension 3). In low dimension, these groups have been widely studied, see SO(2) , SO(3) and SO(4) .
The symmetry group of a square belongs to the family of dihedral groups, D n (abstract group type Dih n), including as many reflections as rotations. The infinite rotational symmetry of the circle implies reflection symmetry as well, but formally the circle group S 1 is distinct from Dih(S 1 ) because the latter explicitly includes the reflections.
This article summarizes the classes of discrete symmetry groups of the Euclidean plane. The symmetry groups are named here by three naming schemes: International notation, orbifold notation, and Coxeter notation. There are three kinds of symmetry groups of the plane: 2 families of rosette groups – 2D point groups; 7 frieze groups – 2D line ...
The odd-dimensional rotation groups do not contain the central inversion and are simple groups. The even-dimensional rotation groups do contain the central inversion −I and have the group C 2 = {I, −I} as their centre. For even n ≥ 6, SO(n) is almost simple in that the factor group SO(n)/C 2 of SO(n) by its centre is a simple group.
The cyclic group C 3 consisting of the rotations by 0°, 120° and 240° acts on the set of the three vertices.. In mathematics, a group action of a group G on a set S is a group homomorphism from G to some group (under function composition) of functions from S to itself.
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