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In group theory, the symmetry group of a geometric object is the group of all transformations under which the object is invariant, endowed with the group operation of composition. Such a transformation is an invertible mapping of the ambient space which takes the object to itself, and which preserves all the relevant structure of the object.
A symmetry on the spacetime is a smooth vector field whose local flow diffeomorphisms preserve some (usually geometrical) feature of the spacetime. The (geometrical) feature may refer to specific tensors (such as the metric, or the energy–momentum tensor) or to other aspects of the spacetime such as its geodesic structure.
The superscript doesn't give any additional information about symmetry elements of the space group, but is instead related to the order in which Schoenflies derived the space groups. This is sometimes supplemented with a symbol of the form Γ x y {\displaystyle \Gamma _{x}^{y}} which specifies the Bravais lattice.
Poincaré symmetry is the full symmetry of special relativity. It includes: translations (displacements) in time and space, forming the abelian Lie group of spacetime translations (P); rotations in space, forming the non-abelian Lie group of three-dimensional rotations (J); boosts, transformations connecting two uniformly moving bodies (K).
The Lorentz group expresses the fundamental symmetry of space and time of all known fundamental laws of nature. In small enough regions of spacetime where gravitational variances are negligible, physical laws are Lorentz invariant in the same manner as special relativity.
The Euclidean group can be seen as the symmetry group of the space itself, and contains the group of symmetries of any figure (subset) of that space. A Euclidean isometry can be direct or indirect , depending on whether it preserves the handedness of figures.
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 groups; 17 wallpaper groups – 2D space groups.
Special positions are left invariant by the identity operation and at least one other operation of the space group. General positions have a site symmetry of the trivial group and all correspond to the same Wyckoff position. Special positions have a non-trivial site symmetry group. For a particular space group, one can check the Wyckoff ...