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Mathematical tools
International Tables for Crystallography (2013). Vol. D, Section 3.2.3, pp. 399-410 [ doi:10.1107/97809553602060000916 ]
... for the double cosets [for derivations and more details, see Janovec (1972)]. The inverse of a double coset is a ... in domain-structure analysis and domain engineering, see e.g. Fuksa & Janovec (1995, 2002)]. 3.2.3.3.6. Orbits of ordered pairs and double ... set of representatives of double cosets in the decomposition (3.2.3.100) (Janovec, 1972). Proposition 3.2.3.35 applies directly to pairs of ...
Intermediate subgroups and partitions of an orbit into suborbits
International Tables for Crystallography (2013). Vol. D, Section 3.2.3.3.5, pp. 408-410 [ doi:10.1107/97809553602060000916 ]
... in domain-structure analysis and domain engineering, see e.g. Fuksa & Janovec (1995, 2002)]. References Aizu, K. (1972). Electrical, mechanical ... crystals. J. Phys. Soc. Jpn, 32, 1287-1301. Fuksa, J. & Janovec, V. (1995). Permutation classification of domain pairs. Ferroelectrics, 172, ...
Orbits and left cosets
International Tables for Crystallography (2013). Vol. D, Section 3.2.3.3.4, p. 408 [ doi:10.1107/97809553602060000916 ]
... A. (1999). Applied finite group actions. Berlin: Springer. Kopský, V. (1983). Algebraic investigations in Landau model of structural phase ...
Orbits
International Tables for Crystallography (2013). Vol. D, Section 3.2.3.3.3, pp. 407-408 [ doi:10.1107/97809553602060000916 ]
Orbits 3.2.3.3.3. Orbits The group action allows one to specify the equivalence relation and the partition of a set into equivalence classes introduced in Section 3.2.3.1 [see (3.2.3.6)]. If G is a group and are two objects of a G-set , then one says that the objects are G-equivalent ...
Stabilizers (isotropy groups)
International Tables for Crystallography (2013). Vol. D, Section 3.2.3.3.2, pp. 406-407 [ doi:10.1107/97809553602060000916 ]
Stabilizers (isotropy groups) 3.2.3.3.2. Stabilizers (isotropy groups) The concept of a stabilizer is closely connected with the notion of the symmetry group of an object. Under the symmetry group F of an object one understands the set of all operations (isometries) that map the object onto itself, i.e. leave this object ...
Group action
International Tables for Crystallography (2013). Vol. D, Section 3.2.3.3.1, pp. 405-406 [ doi:10.1107/97809553602060000916 ]
Group action 3.2.3.3.1. Group action A direct application of the set and group theory to our studies would hardly justify their presentation in the last two sections. However, an appropriate combination of these theories, called group action, forms a very useful tool for examining crystalline materials and domain structures in particular. ...
Action of a group on a set
International Tables for Crystallography (2013). Vol. D, Section 3.2.3.3, pp. 405-410 [ doi:10.1107/97809553602060000916 ]
... in domain-structure analysis and domain engineering, see e.g. Fuksa & Janovec (1995, 2002)]. 3.2.3.3.6. Orbits of ordered pairs and double ... set of representatives of double cosets in the decomposition (3.2.3.100) (Janovec, 1972). Proposition 3.2.3.35 applies directly to pairs of domain ... crystals. J. Phys. Soc. Jpn, 32, 1287-1301. Fuksa, J. & Janovec, V. (1995). Permutation classification of domain pairs. Ferroelectrics, ...
Double cosets
International Tables for Crystallography (2013). Vol. D, Section 3.2.3.2.8, pp. 404-405 [ doi:10.1107/97809553602060000916 ]
... for the double cosets [for derivations and more details, see Janovec (1972)]. The inverse of a double coset is a ... . The theory of groups. New York: The Macmillan Company. Janovec, V. (1972). Group analysis of domains and domain pairs. ...
Halving subgroups and dichromatic (black-and-white) groups
International Tables for Crystallography (2013). Vol. D, Section 3.2.3.2.7, p. 404 [ doi:10.1107/97809553602060000916 ]
Halving subgroups and dichromatic (black-and-white) groups 3.2.3.2.7. Halving subgroups and dichromatic (black-and-white) groups Any subgroup H of a group G of index 2, called a halving subgroup, is a normal subgroup. The decomposition of G into left cosets of H consists of two left cosets, Sometimes it ...
Normal subgroups
International Tables for Crystallography (2013). Vol. D, Section 3.2.3.2.6, p. 403 [ doi:10.1107/97809553602060000916 ]
Normal subgroups 3.2.3.2.6. Normal subgroups Among subgroups of a group, a special role is played by normal subgroups. A subgroup H of G is a normal (invariant, self-conjugate) subgroup of G if and only if it fulfils any of the following conditions: (1) The subgroup H of G has no ...
International Tables for Crystallography (2013). Vol. D, Section 3.2.3, pp. 399-410 [ doi:10.1107/97809553602060000916 ]
... for the double cosets [for derivations and more details, see Janovec (1972)]. The inverse of a double coset is a ... in domain-structure analysis and domain engineering, see e.g. Fuksa & Janovec (1995, 2002)]. 3.2.3.3.6. Orbits of ordered pairs and double ... set of representatives of double cosets in the decomposition (3.2.3.100) (Janovec, 1972). Proposition 3.2.3.35 applies directly to pairs of ...
Intermediate subgroups and partitions of an orbit into suborbits
International Tables for Crystallography (2013). Vol. D, Section 3.2.3.3.5, pp. 408-410 [ doi:10.1107/97809553602060000916 ]
... in domain-structure analysis and domain engineering, see e.g. Fuksa & Janovec (1995, 2002)]. References Aizu, K. (1972). Electrical, mechanical ... crystals. J. Phys. Soc. Jpn, 32, 1287-1301. Fuksa, J. & Janovec, V. (1995). Permutation classification of domain pairs. Ferroelectrics, 172, ...
Orbits and left cosets
International Tables for Crystallography (2013). Vol. D, Section 3.2.3.3.4, p. 408 [ doi:10.1107/97809553602060000916 ]
... A. (1999). Applied finite group actions. Berlin: Springer. Kopský, V. (1983). Algebraic investigations in Landau model of structural phase ...
Orbits
International Tables for Crystallography (2013). Vol. D, Section 3.2.3.3.3, pp. 407-408 [ doi:10.1107/97809553602060000916 ]
Orbits 3.2.3.3.3. Orbits The group action allows one to specify the equivalence relation and the partition of a set into equivalence classes introduced in Section 3.2.3.1 [see (3.2.3.6)]. If G is a group and are two objects of a G-set , then one says that the objects are G-equivalent ...
Stabilizers (isotropy groups)
International Tables for Crystallography (2013). Vol. D, Section 3.2.3.3.2, pp. 406-407 [ doi:10.1107/97809553602060000916 ]
Stabilizers (isotropy groups) 3.2.3.3.2. Stabilizers (isotropy groups) The concept of a stabilizer is closely connected with the notion of the symmetry group of an object. Under the symmetry group F of an object one understands the set of all operations (isometries) that map the object onto itself, i.e. leave this object ...
Group action
International Tables for Crystallography (2013). Vol. D, Section 3.2.3.3.1, pp. 405-406 [ doi:10.1107/97809553602060000916 ]
Group action 3.2.3.3.1. Group action A direct application of the set and group theory to our studies would hardly justify their presentation in the last two sections. However, an appropriate combination of these theories, called group action, forms a very useful tool for examining crystalline materials and domain structures in particular. ...
Action of a group on a set
International Tables for Crystallography (2013). Vol. D, Section 3.2.3.3, pp. 405-410 [ doi:10.1107/97809553602060000916 ]
... in domain-structure analysis and domain engineering, see e.g. Fuksa & Janovec (1995, 2002)]. 3.2.3.3.6. Orbits of ordered pairs and double ... set of representatives of double cosets in the decomposition (3.2.3.100) (Janovec, 1972). Proposition 3.2.3.35 applies directly to pairs of domain ... crystals. J. Phys. Soc. Jpn, 32, 1287-1301. Fuksa, J. & Janovec, V. (1995). Permutation classification of domain pairs. Ferroelectrics, ...
Double cosets
International Tables for Crystallography (2013). Vol. D, Section 3.2.3.2.8, pp. 404-405 [ doi:10.1107/97809553602060000916 ]
... for the double cosets [for derivations and more details, see Janovec (1972)]. The inverse of a double coset is a ... . The theory of groups. New York: The Macmillan Company. Janovec, V. (1972). Group analysis of domains and domain pairs. ...
Halving subgroups and dichromatic (black-and-white) groups
International Tables for Crystallography (2013). Vol. D, Section 3.2.3.2.7, p. 404 [ doi:10.1107/97809553602060000916 ]
Halving subgroups and dichromatic (black-and-white) groups 3.2.3.2.7. Halving subgroups and dichromatic (black-and-white) groups Any subgroup H of a group G of index 2, called a halving subgroup, is a normal subgroup. The decomposition of G into left cosets of H consists of two left cosets, Sometimes it ...
Normal subgroups
International Tables for Crystallography (2013). Vol. D, Section 3.2.3.2.6, p. 403 [ doi:10.1107/97809553602060000916 ]
Normal subgroups 3.2.3.2.6. Normal subgroups Among subgroups of a group, a special role is played by normal subgroups. A subgroup H of G is a normal (invariant, self-conjugate) subgroup of G if and only if it fulfils any of the following conditions: (1) The subgroup H of G has no ...
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