International
Tables for
Crystallography
Volume A
Space-group symmetry
Edited by Th. Hahn
eISBN 978-1-4020-5406-8


© International Union of Crystallography 2006
International Tables for Crystallography (2006). Vol. A, ch. 8.1, pp. 720-725
doi: 10.1107/97809553602060000514

Chapter 8.1. Basic concepts

H. Wondratscheka*

aInstitut für Kristallographie, Universität, D-76128 Karlsruhe, Germany
Correspondence e-mail: hans.wondratschek@physik.uni-karlsruhe.de

Footnotes

1 For this volume, the following conventions for the writing of vectors and matrices have been adopted:
(i) point coordinates and vector coefficients are written as [(n \times 1)] column matrices;
(ii) the vectors of the vector basis are written as a [(1 \times n)] row matrix;
(iii) all running indices are written as subscripts.
It should be mentioned that other conventions are also found in the literature, e.g. interchange of row and column matrices and simultaneous use of subscripts and superscripts for running indices.
2 The reflection [m \equiv \bar{2}] is contained among the rotoinversions. The same restriction is valid for the rotation angle ϕ in two-dimensional space, where [\hbox{tr}({\bi W}) = 2 \cos \varphi] if [\det ({\bi W}) = + 1]. If [\det ({\bi W}) = - 1, \ \hbox{tr}({\bi W}) = 0] always holds and the operation is a reflection m.
3 A method of deriving the possible orders of W in spaces of arbitrary dimension has been described by Hermann (1949)[link].
4 For a rigorous definition of the term symmetry element, see de Wolff et al. (1989[link], 1992[link]) and Flack et al. (2000)[link].
5 A coset decomposition of a group [{\cal G}] is possible with respect to every subgroup [{\cal H}] of [{\cal G}] ; cf. Ledermann (1976)[link]. The number of cosets is called the index [i] of [{\cal H}] in [{\cal G}]. The integer [i] may be finite, as for the coset decomposition of a space group [{\cal G}] with respect to the (infinite) translation group [{\cal T}] or infinite, as for the coset decomposition of a space group [{\cal G}] with respect to a (finite) site-symmetry group [{\cal S}]; cf. Section 8.3.2[link] . If [{\cal G}] is a finite group, a theorem of Lagrange states that the order of [{\cal G}] is the product of the order of [{\cal H}] and the index of [{\cal H}] in [{\cal G}].