International Tables for Crystallography (2011). Vol. A1. ch. 1.2, pp. 7-24   | 1 | 2 |
https://doi.org/10.1107/97809553602060000791

Chapter 1.2. General introduction to the subgroups of space groups

Contents

  • 1.2. General introduction to the subgroups of space groups  (pp. 7-24) | html | pdf | chapter contents |
    Hans Wondratschek
    • 1.2.1. General remarks  (p. 7) | html | pdf |
    • 1.2.2. Mappings and matrices  (pp. 7-10) | html | pdf |
      • 1.2.2.1. Crystallographic symmetry operations  (pp. 7-8) | html | pdf |
      • 1.2.2.2. Coordinate systems and coordinates  (p. 8) | html | pdf |
      • 1.2.2.3. The description of mappings  (pp. 8-9) | html | pdf |
      • 1.2.2.4. Matrix–column pairs and (n + 1) × (n + 1) matrices  (p. 9) | html | pdf |
      • 1.2.2.5. Isometries  (p. 9) | html | pdf |
      • 1.2.2.6. Vectors and vector coefficients  (p. 10) | html | pdf |
      • 1.2.2.7. Origin shift and change of the basis  (p. 10) | html | pdf |
    • 1.2.3. Groups  (pp. 10-12) | html | pdf |
      • 1.2.3.1. Some properties of symmetry groups  (p. 11) | html | pdf |
      • 1.2.3.2. Group isomorphism and homomorphism  (pp. 11-12) | html | pdf |
    • 1.2.4. Subgroups  (pp. 12-14) | html | pdf |
      • 1.2.4.1. Definition  (p. 12) | html | pdf |
      • 1.2.4.2. Coset decomposition and normal subgroups  (pp. 12-13) | html | pdf |
      • 1.2.4.3. Conjugate elements and conjugate subgroups  (p. 13) | html | pdf |
      • 1.2.4.4. Factor groups and homomorphism  (p. 13) | html | pdf |
      • 1.2.4.5. Normalizers  (pp. 13-14) | html | pdf |
    • 1.2.5. Space groups  (pp. 14-18) | html | pdf |
      • 1.2.5.1. Space groups and their description  (pp. 14-15) | html | pdf |
      • 1.2.5.2. Classifications of space groups  (p. 15) | html | pdf |
      • 1.2.5.3. Space groups and space-group types  (pp. 15-16) | html | pdf |
      • 1.2.5.4. Point groups and crystal classes  (pp. 16-17) | html | pdf |
      • 1.2.5.5. Crystal systems and crystal families  (pp. 17-18) | html | pdf |
    • 1.2.6. Types of subgroups of space groups  (pp. 18-20) | html | pdf |
      • 1.2.6.1. Introductory remarks  (p. 18) | html | pdf |
      • 1.2.6.2. Definitions and examples  (pp. 18-19) | html | pdf |
      • 1.2.6.3. The role of normalizers for group–subgroup pairs of space groups  (pp. 19-20) | html | pdf |
    • 1.2.7. Application to domain structures  (pp. 20-23) | html | pdf |
      • 1.2.7.1. Introductory remarks  (p. 20) | html | pdf |
      • 1.2.7.2. Domain states, symmetry states and directional states  (pp. 20-22) | html | pdf |
      • 1.2.7.3. Examples  (pp. 22-23) | html | pdf |
    • 1.2.8. Lemmata on subgroups of space groups  (pp. 23-24) | html | pdf |
      • 1.2.8.1. General lemmata  (p. 23) | html | pdf |
      • 1.2.8.2. Lemmata on maximal subgroups  (pp. 23-24) | html | pdf |
    • References | html | pdf |
    • Figures
      • Fig. 1.2.7.1. Group–subgroup relations between the high (HT) and low temperature (LT) forms of gadolinium molybdate (Bärnighausen tree as explained in Section 1.6.3[link] )  (p. 23) | html | pdf |
    • Tables
      • Table 1.2.3.1. Multiplication table of a group  (p. 11) | html | pdf |