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[link] , it is convenient to use the `short' symbols mm, 32, 3m, [\bar3m], [\bar4m], and [\bar6m] 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[link].
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[link], 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[link], p. 255).

  • 3. A space group is called symmorphic if the coset representatives [{\sf W_j}] can be chosen in such a way that they leave one common point fixed (Wondratschek, 2005[link], Section 8.1.6[link] ).

Even in context, these are pretty opaque.