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Results for DC.creator="H." AND DC.creator="Wondratschek" in section 1.2.4 of volume A1 |
Normalizers
International Tables for Crystallography (2011). Vol. A1, Section 1.2.4.5, pp. 13-14 [ doi:10.1107/97809553602060000791 ]
Normalizers 1.2.4.5. Normalizers The concept of the normalizer of a group in a group is very useful for the considerations of the following sections. The length of the conjugacy class of in is determined by this normalizer. Let and . Then holds because is a group. If , then for any . ...
Factor groups and homomorphism
International Tables for Crystallography (2011). Vol. A1, Section 1.2.4.4, p. 13 [ doi:10.1107/97809553602060000791 ]
Factor groups and homomorphism 1.2.4.4. Factor groups and homomorphism For the following definition, the `product of sets of group elements' will be used: Definition 1.2.4.4.1.Let be a group and , be two arbitrary sets of its elements which are not necessarily groups themselves. Then the product of and is the set of ...
Conjugate elements and conjugate subgroups
International Tables for Crystallography (2011). Vol. A1, Section 1.2.4.3, p. 13 [ doi:10.1107/97809553602060000791 ]
Conjugate elements and conjugate subgroups 1.2.4.3. Conjugate elements and conjugate subgroups In a coset decomposition, the set of all elements of the group is partitioned into cosets which form classes in the mathematical sense of the word, i.e. each element of belongs to exactly one coset. Another equally important partition of ...
Coset decomposition and normal subgroups
International Tables for Crystallography (2011). Vol. A1, Section 1.2.4.2, pp. 12-13 [ doi:10.1107/97809553602060000791 ]
Coset decomposition and normal subgroups 1.2.4.2. Coset decomposition and normal subgroups Let be a subgroup of of order . Because is a proper subgroup of there must be elements that are not elements of . Let be one of them. Then the set of elements 4 is a subset of elements ...
Definition
International Tables for Crystallography (2011). Vol. A1, Section 1.2.4.1, p. 12 [ doi:10.1107/97809553602060000791 ]
Definition 1.2.4.1. Definition There may be sets of elements that do not constitute the full group but nevertheless fulfil the group postulates for themselves. Definition 1.2.4.1.1.A subset of elements of a group is called a subgroup of if it fulfils the group postulates with respect to the law of composition of ...
Subgroups
International Tables for Crystallography (2011). Vol. A1, Section 1.2.4, pp. 12-14 [ doi:10.1107/97809553602060000791 ]
Subgroups 1.2.4. Subgroups 1.2.4.1. Definition | | There may be sets of elements that do not constitute the full group but nevertheless fulfil the group postulates for themselves. Definition 1.2.4.1.1.A subset of elements of a group is called a subgroup of if it fulfils the group postulates with respect to the law of ...
International Tables for Crystallography (2011). Vol. A1, Section 1.2.4.5, pp. 13-14 [ doi:10.1107/97809553602060000791 ]
Normalizers 1.2.4.5. Normalizers The concept of the normalizer of a group in a group is very useful for the considerations of the following sections. The length of the conjugacy class of in is determined by this normalizer. Let and . Then holds because is a group. If , then for any . ...
Factor groups and homomorphism
International Tables for Crystallography (2011). Vol. A1, Section 1.2.4.4, p. 13 [ doi:10.1107/97809553602060000791 ]
Factor groups and homomorphism 1.2.4.4. Factor groups and homomorphism For the following definition, the `product of sets of group elements' will be used: Definition 1.2.4.4.1.Let be a group and , be two arbitrary sets of its elements which are not necessarily groups themselves. Then the product of and is the set of ...
Conjugate elements and conjugate subgroups
International Tables for Crystallography (2011). Vol. A1, Section 1.2.4.3, p. 13 [ doi:10.1107/97809553602060000791 ]
Conjugate elements and conjugate subgroups 1.2.4.3. Conjugate elements and conjugate subgroups In a coset decomposition, the set of all elements of the group is partitioned into cosets which form classes in the mathematical sense of the word, i.e. each element of belongs to exactly one coset. Another equally important partition of ...
Coset decomposition and normal subgroups
International Tables for Crystallography (2011). Vol. A1, Section 1.2.4.2, pp. 12-13 [ doi:10.1107/97809553602060000791 ]
Coset decomposition and normal subgroups 1.2.4.2. Coset decomposition and normal subgroups Let be a subgroup of of order . Because is a proper subgroup of there must be elements that are not elements of . Let be one of them. Then the set of elements 4 is a subset of elements ...
Definition
International Tables for Crystallography (2011). Vol. A1, Section 1.2.4.1, p. 12 [ doi:10.1107/97809553602060000791 ]
Definition 1.2.4.1. Definition There may be sets of elements that do not constitute the full group but nevertheless fulfil the group postulates for themselves. Definition 1.2.4.1.1.A subset of elements of a group is called a subgroup of if it fulfils the group postulates with respect to the law of composition of ...
Subgroups
International Tables for Crystallography (2011). Vol. A1, Section 1.2.4, pp. 12-14 [ doi:10.1107/97809553602060000791 ]
Subgroups 1.2.4. Subgroups 1.2.4.1. Definition | | There may be sets of elements that do not constitute the full group but nevertheless fulfil the group postulates for themselves. Definition 1.2.4.1.1.A subset of elements of a group is called a subgroup of if it fulfils the group postulates with respect to the law of ...
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