Table of contents

  1. Front Matter
    Pages i-xii
  2. Space groups and their subgroups

    1. M. I. Aroyo, U. Müller, H. Wondratschek
      Pages 2-5
    2. U. Müller
      Pages 24-26
  3. Maximal subgroups of the plane groups and space groups

    1. H. Wondratschek, M. I. Aroyo
      Pages 42-58
    2. Y. Billiet, M. I. Aroyo, H. Wondratschek
      Pages 59-76
    3. Y. Billiet, M. I. Aroyo, H. Wondratschek
      Pages 77-394
    4. V. Gramlich, H. Wondratschek
      Pages 395-414
    5. V. Gramlich, H. Wondratschek
      Pages 415-425
  4. Relations between the Wyckoff positions

    1. U. Müller
      Pages 428-433
  5. Back Matter
    Pages 723-731

About this book


This volume presents a systematic treatment of the maximal subgroups and minimal supergroups of the crystallographic plane groups and space groups. It is an extension of and a supplement to Volume A, Space-group symmetry, in which only basic data for sub- and supergroups are provided.

Group-subgroup relations, apart from their theoretical interest, are the basis of a number of important applications in crystallographic research:

(1) In solid-state phase transitions there often exists a group-subgroup relation between the symmetry groups of the two phases. According to Landau theory, this is in fact mandatory for displacive (continuous, second-order) phase transitions. Group-subgroup relations are also indispensable in cases where the symmetry groups of the two phases are not directly related but share a common subgroup or supergroup.

(2) Group-subgroup relations provide a concise and powerful tool for revealing and elucidating relations between crystal structures. They can thus help to keep up with the ever-increasing amount of crystal-structure data. Their application requires knowledge of the relations of the Wyckoff positions of group-subgroup related structures.

(3) Group-subgroup relations are of great importance in the study of twinned crystals, domain structures and domain boundaries.

(4) These relations can even help to identify errors in space-group assignment and crystal-structure determination.

(5) Subgroups of space groups provide a valuable approach to teaching crystallographic symmetry.

Volume A1 consists of three parts:

Part 1 presents an introduction to the theory of space groups at various levels and with many examples. It includes a chapter on the mathematical theory of subgroups.

Part 2 gives for each plane group and space group a complete listing of all maximal subgroups and minimal supergroups. The treatment includes the generators of each subgroup as well as any necessary changes of the coordinate system. Maximal isomorphic subgroups are given in parameterized form as infinite series because of the infinite number for each group. A special feature of the presentation is graphs that illustrate the group-subgroup relations.

Part 3 lists the relations between the Wyckoff positions of every space group and its subgroups. Again, the infinite number of maximal isomorphic subgroups of each space group are covered by parameterized series. These data for Wyckoff positions are presented here for the first time.

Audience: The volume is a valuable addition to the library of scientists engaged in crystal-structure determination, crystal physics or crystal chemistry. It is essential for those interested in phase transitions, the systematic compilation of crystal structures, twinning phenomena and related fields of crystallographic research.


chemistry crystal crystal chemistry crystal structure crystallography phase transition structure symmetry

Editors and affiliations

  • H. Wondratschek
    • 1
  • U. Müller
    • 2
  1. 1.Institut für KristallographieUniversitätKarlsruheGermany
  2. 2.Fachbereich ChemiePhilipps-UniversitätMarburgGermany

Bibliographic information

  • Book Title International Tables for Crystallography Volume A1: Symmetry relations between space groups
  • Editors H. Wondratschek
    U. Müller
  • Series Title International Tables for Crystallography
  • Series Abbreviated Title Int. Tables Crystallogr.
  • DOI
  • Copyright Information International Union of Crystallography 2004
  • Publisher Name Springer, Dordrecht
  • eBook Packages Springer Book Archive
  • eBook ISBN 978-1-4020-5413-6
  • Series E-ISSN 1574-8707
  • Edition Number 1
  • Number of Pages , 0
  • Number of Illustrations 0 b/w illustrations, 0 illustrations in colour
  • Additional Information Jointly published with the IUCR
  • Topics Crystallography and Scattering Methods
    Analytical Chemistry
    Physical Chemistry
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