International
Tables for
Crystallography
Volume C
Mathematical, physical and chemical tables
Edited by E. Prince

International Tables for Crystallography (2006). Vol. C, ch. 1.4, pp. 15-22
https://doi.org/10.1107/97809553602060000575

## Chapter 1.4. Arithmetic crystal classes and symmorphic space groups

A. J. C. Wilsona

aSt John's College, Cambridge CB2 1TP, England

## Footnotes

Deceased.

1 Here and in Chapter 9.7 , it is convenient to use the short' symbols mm, 32, 3m, , , and instead of mm2, 321, etc., whenever it is desired to emphasize that no implication about orientation is intended.
2 In the arithmetic crystal class 2mmC, two conventions concerning the nomenclature of axes conflict. The first is that, if only one face of the Bravais lattice is centred, the c axis is chosen perpendicular to that face. The second is that, if there is one axis of symmetry uniquely different from any others, that axis is to be chosen as b in the monoclinic system and as c in the remaining systems. The second convention is usually regarded as the more important, and the standard setting' of 2mmC is mm2A. Both settings are listed in Table 1.4.2.1.
3 Three examples of informative definitions are:
 1. The space group corresponding to the zero solution of the Frobenius congruences is called a symmorphic space group (Engel, 1986, p. 155). 2. A space group F is called symmorphic if one of its finite subgroups (and therefore an infinity of them) is of an order equal to the order of the point group Rr (Opechowski, 1986, p. 255). 3. A space group is called symmorphic if the coset representatives can be chosen in such a way that they leave one common point fixed (Wondratschek, 2005, Section 8.1.6 ).
Even in context, these are pretty opaque.