Normalizers of point groups

Koch, E.; Fischer, W.

doi:10.1107/97809553602060000536

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

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International Tables for Crystallography (2006). Vol. A. ch. 15.4, pp. 904-905
https://doi.org/10.1107/97809553602060000536

Chapter 15.4. Normalizers of point groups

E. Koch^a ^* and W. Fischer^a

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

In Chapter 15.4, the normalizers of the two-dimensional point groups with respect to the full isometry group of the circle and of the three-dimensional point groups with respect to the full isometry group of the sphere are tabulated.

Keywords: point groups; normalizers; isometry group of a circle; isometry group of a sphere.

Normalizers with respect to the Euclidean or affine group may be defined for any group of isometries (cf. Gubler, 1982a ,b ). For a point group, however, it seems inadequate to use a supergroup that contains transformations that do not map a fixed point of that point group onto itself. Appropriate supergroups for the definition of normalizers of point groups are the full isometry groups of the sphere, $[m\overline{\infty}]$ , and of the circle, ∞m, in three-dimensional and two-dimensional space (cf. Galiulin, 1978 ).

These normalizers are listed in Tables 15.4.1.1 and 15.4.1.2 . It has to be noticed that the normalizer of a crystallographic point group may contain continuous rotations, i.e. rotations with infinitesimal rotation angle, or noncrystallographic rotations ( $[\infty m]$ ; $[m\overline{\infty}, \infty/mm, 8mm, 12mm]$ ; [8/mmm, 12/mmm] ). In analogy to space groups, these normalizers define equivalence relationships on the `Wyckoff positions' of the point groups (cf. Section 10.1.2 ). They give also the relation between the different but equivalent morphological descriptions of a crystal.

Table 15.4.1.1| top | pdf |
Normalizers of the two-dimensional point groups with respect to the full isometry group of the circle

The upper part refers to the crystallographic, the lower part to the noncrystallographic point groups as listed in Table 10.1.4.1 .

Normalizer	Point groups
∞m	1, 2, 4, 3, 6
12mm	6mm
8mm	4mm
6mm	3m
4mm	2mm
2mm	m
∞m	$[n, \infty, \infty m]$
(2n)mm	nmm, nm

Table 15.4.1.2| top | pdf |
Normalizers of the three-dimensional point groups with respect to the full isometry group of the sphere

The upper part refers to the crystallographic, the lower part to the noncrystallographic point groups as listed in Table 10.1.4.2 .

Normalizer	Point groups
$[m\overline{\infty}]$	$[1, \overline{1}]$
$[m\overline{3}m]$	$[222, mmm, 23, m\overline{3}, 432, \overline{4}3m, m\overline{3}m]$
$[\infty/mm]$	$[2, m, 2/m, 4, \overline{4}, 4/m, 3, \overline{3}, 6, \overline{6}, 6/m]$


	$[32, 3m, \overline{3}m, \overline{6}2m]$
	$[mm2, \overline{4}2m]$
$[m\overline{\infty}]$	$[2\infty, m\overline{\infty}]$
$[m\overline{35}]$	$[235, m\overline{35}]$
$[\infty/mm]$	$[n, \overline{n}, n/m, \infty, \infty/m, \infty 2, \infty m, \infty/mm]$
	$[n22, nmm, n/mmm, n2, nm, \overline{n}m]$
	$[\overline{n}2m]$

References

Galiulin, R. V. (1978). Holohedral varieties of simple forms of crystals. Sov. Phys. Crystallogr. 23, 635–641.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

International Tables for Crystallography (2006). Vol. A. ch. 15.4, pp. 904-905
https://doi.org/10.1107/97809553602060000536