International
Tables for
Crystallography
Volume A1
Symmetry relations between space groups
Edited by Hans Wondratschek and Ulrich Müller

International Tables for Crystallography (2006). Vol. A1, ch. 1.5, p. 29   | 1 | 2 |

## Section 1.5.1. Introduction

Gabriele Nebea*

aAbteilung Reine Mathematik, Universität Ulm, D-89069 Ulm, Germany
Correspondence e-mail: nebe@mathematik.uni-ulm.de

### 1.5.1. Introduction

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This chapter gives a brief introduction to the mathematics involved in the determination of the subgroups of space groups. To achieve this we have to detach ourselves from the geometric point of view in crystallography and introduce more abstract algebraic structures, such as coordinates, which are well known in crystallography and permit the formalization of symmetry operations, and also the abstract notion of a group, which allows us to apply general theorems to the concrete situation of (three-dimensional) space groups.

This algebraic point of view has the following advantages:

 (1) Geometric problems can be treated by algebraic calculations. These calculations can be dealt with by well established procedures. In particular, the use of computers and advanced programs enables one to solve even difficult problems in a comparatively short time. (2) The mappings form groups in the mathematical sense of the word. This means that the very powerful methods of group theory may be applied successfully. (3) The procedures for the solution may be developed to a great extent independently of the dimension of the space.

In Section 1.5.2, a basis is laid down which gives the reader an understanding of the algebraic point of view of the crystal space (or point space) and special mappings of this space onto itself. The set of these mappings is an example of a group. For a closer connection to crystallography, the reader may consult Section 8.1.1 of IT A (2005) or the book by Hahn & Wondratschek (1994).

Section 1.5.3 gives an introduction to abstract groups and states the important theorems of group theory that will be applied in Section 1.5.4 to the most important groups in crystallography, the space groups. In particular, Section 1.5.4 treats maximal subgroups of space groups which have a special structure by the theorem of Hermann. In Section 1.5.5, we come back to abstract group theory stating general facts about maximal subgroups of groups. These general theorems allow us to calculate the possible indices of maximal subgroups of three-dimensional space groups in Section 1.5.6. The last section, Section 1.5.7, deals with the very subtle question of when these maximal subgroups of a space group are isomorphic to this space group.

### References

Hahn, Th. & Wondratschek, H. (1994). Symmetry of crystals. Introduction to International Tables for Crystallography, Vol. A. Sofia: Heron Press.
International Tables for Crystallography (2005). Vol. A, Space-group symmetry, edited by Th. Hahn, 5th ed. Heidelberg: Springer. (Abbreviated IT A.)