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

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.