Definitions

Koch, E.; Fischer, W.; Müller, U.

doi:10.1107/97809553602060000533

International
Tables for
Crystallography
Volume A
Space-group symmetry
Edited by Th. Hahn

pdf | chapter contents | chapter index | related articles

International Tables for Crystallography (2006). Vol. A. ch. 15.1, p. 878

Section 15.1.2. Definitions

E. Koch,^a ^* W. Fischer^a and U. Müller^b

^a Institut für Mineralogie, Petrologie und Kristallographie, Philipps-Universität, D-35032 Marburg, Germany, and ^bFachbereich Chemie, Philipps-Universität, D-35032 Marburg, Germany
Correspondence e-mail: kochelke@mailer.uni-marburg.de

15.1.2. Definitions

| top | pdf |

Any pair, consisting of a group $[{\cal G}]$ and one of its supergroups $[{\cal S}]$ , is uniquely related to a third intermediate group $[{\cal N}\!_{{\cal S}}({\cal G})]$ , called the normalizer of $[{\cal G}]$ with respect to $[{\cal S}]$ . $[{\cal N}\!_{{\cal S}}({\cal G})]$ is defined as the set of all elements $[{\bf S} \in {\cal S}]$ that map $[{\cal G}]$ onto itself by conjugation (cf. Section 8.3.6 ). $[{\cal N}\!_{{\cal S}}({\cal G}) := \{ {\bf S} \in {\cal S} |\ {\bf S}^{-1} {\cal G}\ {\bf S} = {\cal G}\}.]$ The normalizer $[{\cal N}\!_{{\cal S}}({\cal G})]$ may coincide either with $[{\cal G}]$ or with $[{\cal S}]$ or it may be a proper intermediate group. In any case, $[{\cal G}]$ is a normal subgroup of its normalizer.

For most crystallographic problems, two kinds of normalizers are of special interest: (i) the normalizer of a space group (plane group) $[{\cal G}]$ with respect to the group $[{\cal E}]$ of all Euclidean mappings (motions, isometries) in $[E^{3}\; (E^{2})]$ , called the Euclidean normalizer of $[{\cal G}]$ $[{\cal N}\!_{{\cal E}}({\cal G}) := \{{\bf S} \in {\cal E} |\ {\bf S}^{-1} {\cal G}\ {\bf S} = {\cal G}\};]$ (ii) the normalizer of a space group (plane group) $[{\cal G}]$ with respect to the group $[{\cal A}]$ of all affine mappings in $[E^{3}\; (E^{2})]$ , called the affine normalizer of $[{\cal G}]$ $[{\cal N}\!_{{\cal A}}({\cal G}) := \{ {\bf S} \in {\cal A} |\ {\bf S}^{-1} {\cal G}\ {\bf S} = {\cal G}\}.]$

The Euclidean normalizers of the space groups were first derived by Hirshfeld (1968) under the name Cheshire groups. They have been tabulated in more detail by Gubler (1982a,b) and Fischer & Koch (1983). The Euclidean normalizers of triclinic and monoclinic space groups with specialized metric have been determined by Koch & Müller (1990). The affine normalizers of the space groups have been listed by Burzlaff & Zimmermann (1980), Billiet et al. (1982) and Gubler (1982a,b). They have also been used for the derivation of Wyckoff sets and the definition of lattice complexes by Koch & Fischer (1975), even though there the automorphism groups of the space groups were tabulated instead of their affine normalizers.

References

Billiet, Y., Burzlaff, H. & Zimmermann, H. (1982). Comment on the paper of H. Burzlaff and H. Zimmermann. `On the choice of origin in the description of space groups'. Z. Kristallogr. 160, 155–157.Google Scholar

Burzlaff, H. & Zimmermann, H. (1980). On the choice of origin in the description of space groups. Z. Kristallogr. 153, 151–179.Google Scholar

Fischer, W. & Koch, E. (1983). On the equivalence of point configurations due to Euclidean normalizers (Cheshire groups) of space groups. Acta Cryst. A39, 907–915.Google Scholar

Gubler, M. (1982a). Über die Symmetrien der Symmetriegruppen: Automorphismengruppen, Normalisatorgruppen und charakteristische Untergruppen von Symmetriegruppen, insbesondere der kristallographischen Punkt- und Raumgruppen. Dissertation, University of Zürich, Switzerland.Google Scholar

Gubler, M. (1982b). Normalizer groups and automorphism groups of symmetry groups. Z. Kristallogr. 158, 1–26.Google Scholar

Hirshfeld, F. L. (1968). Symmetry in the generation of trial structures. Acta Cryst. A24, 301–311.Google Scholar

Koch, E. & Fischer, W. (1975). Automorphismengruppen von Raumgruppen und die Zuordnung von Punktlagen zu Konfigurationslagen. Acta Cryst. A31, 88–95.Google Scholar

Koch, E. & Müller, U. (1990). Euklidische Normalisatoren für trikline und monokline Raumgruppen bei spezieller Metrik des Translationengitters. Acta Cryst. A46, 826–831.Google Scholar

International Tables for Crystallography (2006). Vol. A. ch. 15.1, p. 878