International
Tables for Crystallography Volume A1 Symmetry relations between space groups Edited by Hans Wondratschek and Ulrich Müller © International Union of Crystallography 2011 
International Tables for Crystallography (2011). Vol. A1, Scope of this volume.
Scope of this volume^{a}Departamento de Física de la Materia Condensada, Facultad de Ciencia y Tecnología, Universidad del País Vasco, Apartado 644, E48080 Bilbao, Spain,^{b}Fachbereich Chemie, PhilippsUniversität, D35032 Marburg, Germany, and ^{c}Institut für Kristallographie, Universität, D76128 Karlsruhe, Germany 
Group–subgroup relations between space groups, the primary subject of this volume, are an important tool in crystallographic, physical and chemical investigations of solids. These relations are complemented by the corresponding relations between the Wyckoff positions of the group–subgroup pairs.
The basis for these tables was laid by the pioneering papers of Carl Hermann in the late 1920s. Some subgroup data were made available in Internationale Tabellen zur Bestimmung von Kristallstrukturen (1935), together with a graph displaying the symmetry relations between the crystallographic point groups.
Since then, the vast number of crystal structures determined and improvements in experimental physical methods have directed the interest of crystallographers, physicists and chemists to the problems of structure classification and of phase transitions. Methods of computational mathematics have been developed and applied to the problems of crystallographic group theory, among them to the group–subgroup relations.
When the new series International Tables for Crystallography began to appear in 1983, the subgroup data that were then available were included in Volume A . However, these data were incomplete and their description was only that which was available in the late 1970s. This is still the case in the present (fifth) edition of Volume A.
The subgroup data for the space groups are now complete and form the basis of this volume. After introductory chapters on grouptheoretical aspects of space groups, on group–subgroup relations and on the underlying mathematical background, this volume provides the reader with:
In this second edition all misprints and errors found up to now have been corrected and the number of illustrating examples has been increased.
In addition, a few changes and extensions have been introduced to facilitate the use of the volume and to extend its range:

The data in this volume are indispensable for a thorough analysis of phase transitions that do not involve drastic structural changes: the group–subgroup relations indicate the possible symmetry breaks that can occur during a phase transition and are essential for determining the symmetry of the driving mechanism and the related symmetry of the resulting phase. The group–subgroup graphs describing the symmetry breaks provide information on the possible symmetry modes taking part in the transition and allow a detailed analysis of domain structures and twins.