## International Tables for Crystallography

## Volume A1: Symmetry relations between space groups

First online edition (2006) ISBN: 978-1-4020-2355-2 doi: 10.1107/97809553602060000101

### Edited by H. Wondratschek and U. Müller

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Volume A1 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.

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. Graphs that illustrate the group-subgroup relations are also presented.
- 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.

Volume A1 is a valuable resource for 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.

Volume A1 was reviewed by R. Gould (*Crystallography News*, No. 92,
March 2005, p. 28).