Normalizers of space groups: A useful tool in crystal-structure description, comparison and determination

Abstract

Abstract After an illustrative example for the ‘symmetry of symmetry’ the group-theoretical concept of normalizers is introduced by a series of definitions. Subsequently, this concept is applied to space groups for which Euclidean and affine normalizers are discussed. Their implications on point configurations, Wyckoff positions and coordinate descriptions of crystal structures are explained. The derivation of all equivalent descriptions of a crystal structure with the aid of the tables of normalizers given in the International Tables for Crystallography, Vol. A is elucidated by several examples.