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Introduction
Souvignier, B.  International Tables for Crystallography (2016). Vol. A, Section 1.3.1, p. 22 [ doi:10.1107/97809553602060000921 ]
Introduction 1.3.1. Introduction We recall from Chapter 1.2 that an isometry is a mapping of the point space which preserves distances and angles. From the mathematical viewpoint, is an affine space in which two points differ by a unique vector in the underlying vector space . The crucial difference between these ...

A general introduction to space groups
Souvignier, B.  International Tables for Crystallography (2016). Vol. A, ch. 1.3, pp. 22-41 [ doi:10.1107/97809553602060000921 ]
... lattice basis. It is customary to choose basis vectors a, b, c which define a right-handed coordinate system, i.e. such that the matrix with columns a, b, c has a positive determinant. Definition For a lattice with ... cells for a rectangular lattice (a) and an oblique lattice (b). It should be noted that the attribute `primitive' ...

Conjugation, normalizers
Souvignier, B.  International Tables for Crystallography (2016). Vol. A, Section 1.1.8, pp. 10-11 [ doi:10.1107/97809553602060000919 ]
... twofold rotation around the linemaps the a axis to the b axis and vice versa, therefore the symmetry operation conjugates to a fourfold rotation with the line along the b axis as geometric element. Since the positive part of the ... a axis is mapped to the positive part of the b axis and conjugation also preserves the handedness of a ...

Group actions
Souvignier, B.  International Tables for Crystallography (2016). Vol. A, Section 1.1.7, pp. 9-10 [ doi:10.1107/97809553602060000919 ]
... i.e. is equivalent to itself: this is easily seen since ; (b) it is symmetric, i.e. if is equivalent to , then is ...

Homomorphisms, isomorphisms
Souvignier, B.  International Tables for Crystallography (2016). Vol. A, Section 1.1.6, pp. 7-9 [ doi:10.1107/97809553602060000919 ]
... that a mapping from a set A to a set B associates to each an element , denoted by and called the ...

Normal subgroups, factor groups
Souvignier, B.  International Tables for Crystallography (2016). Vol. A, Section 1.1.5, pp. 6-7 [ doi:10.1107/97809553602060000919 ]
Normal subgroups, factor groups 1.1.5. Normal subgroups, factor groups In general, the left and right cosets of a subgroup differ, for example in the symmetry group 3m of an equilateral triangle the left coset decomposition with respect to the subgroup is whereas the right coset decomposition is For particular subgroups, however ...

Cosets
Souvignier, B.  International Tables for Crystallography (2016). Vol. A, Section 1.1.4, pp. 5-6 [ doi:10.1107/97809553602060000919 ]
Cosets 1.1.4. Cosets A subgroup allows us to partition a group into disjoint subsets of the same size, called cosets. Definition.Let be a subgroup of . Then for the set is called the left coset of with representative . Analogously, the right coset with representative is defined as The coset is ...

Subgroups
Souvignier, B.  International Tables for Crystallography (2016). Vol. A, Section 1.1.3, pp. 4-5 [ doi:10.1107/97809553602060000919 ]
Subgroups 1.1.3. Subgroups The group of symmetry operations of a crystal pattern may alter if the crystal undergoes a phase transition. Often, some symmetries are preserved, while others are lost, i.e. symmetry breaking takes place. The symmetry operations that are preserved form a subset of the original symmetry group which is ...

Basic properties of groups
Souvignier, B.  International Tables for Crystallography (2016). Vol. A, Section 1.1.2, pp. 2-4 [ doi:10.1107/97809553602060000919 ]
Basic properties of groups 1.1.2. Basic properties of groups Although groups occur in innumerable contexts, their basic properties are very simple and are captured by the following definition. Definition.Let be a set of elements on which a binary operation is defined which assigns to each pair of elements the composition . ...

Introduction
Souvignier, B.  International Tables for Crystallography (2016). Vol. A, Section 1.1.1, p. 2 [ doi:10.1107/97809553602060000919 ]
... two operations (called their composition) is again a symmetry operation. (b) Every symmetry operation can be reversed by simply moving every ...

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