International
Tables for Crystallography Volume D Physical properties of crystals Edited by A. Authier © International Union of Crystallography 2006 |
International Tables for Crystallography (2006). Vol. D. ch. 3.2, p. 388
Section 3.2.3.3.4. Orbits and left cosets
a
Department of Physics, Technical University of Liberec, Hálkova 6, 461 17 Liberec 1, Czech Republic,bInstitut für Kristallographie, Rheinisch–Westfälische Technische Hochschule, D-52056 Aachen, Germany, and cMineralogisch-Petrologisches Institut, Universität Bonn, D-53113 Bonn, Germany |
Proposition 3.2.3.23 . Let G be a finite group, a G-set and the stabilizer of an object of the set . The objects of the orbit and the left cosets of the decomposition of G, are in a one-to-one correspondence,
(See e.g. Kerber, 1991, 1999; Kopský, 1983; Lang, 1965.) The derivation of the bijection (3.2.3.69) consists of two parts:
We note that the orbit depends on the stabilizer of the object and not on the `eigensymmetry' of .
From Proposition 3.2.3.23 follow two corollaries:
Corollary 3.2.3.24 . The order n of the orbit equals the index of the stabilizer in G, where the last part of the equation applies to point groups only.
Corollary 3.2.3.25 . All objects of the orbit can be generated by successive application of representatives of all left cosets in the decomposition of G [see (3.2.3.68)] to the object , . The orbit can therefore be expressed explicitly aswhere the operations (left transversal to in G) are the representatives of left cosets in the decomposition (3.2.3.68).
Example [oP] 3.2.3.26 . The number of equivalent points of the point form (G orbit of the point X) is called a multiplicity of this point, The multiplicity of a point of general position equals the order of the group G, since in this case , a trivial group. Then points of the orbit and the operations of G are in a one-to-one correspondence. The multiplicity of a point of special position is smaller than the order , , and the operations of G and the points of the orbit are in a many-to-one correspondence. Points of a stratum have the same multiplicity; one can, therefore, talk about the multiplicity of the Wyckoff position [see IT A (2005)]. If G is a space group, the point orbit has to be confined to the volume of the primitive unit cell (Wondratschek, 1995).
Example [oC] 3.2.3.27 . Corollaries 3.2.3.24 and 3.2.3.25 applied to domain states represent the basic relations of domain-structure analysis. According to (3.2.3.71), the index n of the stabilizer in the parent group G gives the number of domain states in the orbit and the relations (3.2.3.72) and (3.2.3.68) give a recipe for constructing domain states of this orbit.
Example [oT] 3.2.3.28 . If is a principal tensor parameter associated with the symmetry descent , then there is a one-to-one correspondence between the elements of the orbit of single domain states and the elements of the orbit of the principal order parameter (points) (see Example [oT] 3.2.3.21), Therefore, single domain states of the orbit can be represented by the principal tensor parameter of the orbit .
Example [oS] 3.2.3.29 . Consider a subgroup of a group G. Since the stabilizer of in G is the normalizer (see Example [sS] 3.2.3.17), the number m of conjugate subgroups is, according to (3.2.3.71), where the last part of the equation applies to point groups only. The orbit of conjugate subgroups is where the operations are the representatives of left cosets in the decomposition
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