International
Tables for
Crystallography
Volume D
Physical properties of crystals
Edited by A. Authier

© International Union of Crystallography 2006
International Tables for Crystallography (2006). Vol. D. ch. 3.2, p. 383

Section 3.2.3.2.5. Normalizers

V. Janovec,a* Th. Hahnb and H. Klapperc

a Department of Physics, Technical University of Liberec, Hálkova 6, 461 17 Liberec 1, Czech Republic,bInstitut für Kristallographie, Rheinisch–Westfälische Technische Hochschule, D-52056 Aachen, Germany, and cMineralogisch-Petrologisches Institut, Universität Bonn, D-53113 Bonn, Germany
Correspondence e-mail:  janovec@fzu.cz

3.2.3.2.5. Normalizers

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The collection of all elements g that fulfil the relation [gF_{i}g^{-1}=F_{i}, \ \ g \in G, \eqno(3.2.3.28)]constitutes a group denoted by [N_{G}(F_{i})] and is called the normalizer of [F_{i}] in G. The normalizer [N_{G}(F_{i})] is a subgroup of G and a supergroup of [F_i], [F_i\subseteq N_{G}(F_{i})\subseteq G. \eqno(3.2.3.29)]

The normalizer [N_{G}(F_{i})] determines the subgroups conjugate to [F_i] under G (see Example 3.2.3.10[link]). The number m of subgroups conjugate to a subgroup [F_{i}] under G equals the index of [N_{G}(F_{i})] in G: [ m=[G:N_{G}(F_i)]=|G|:|N_{G}(F_i)|, \eqno(3.2.3.30)] where the last equation holds for finite G and [F_i].

Normalizers of the subgroups of crystallographic point groups are available in Table 3.4.2.7[link] and in the software GI[\star]KoBo-1 under Subgroups\View\Twinning Group.








































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