Conjugate subgroups, normalizers and equivalent descriptions of crystal structures
Conjugate subgroups, normalizers and equivalent descriptions of crystal structures
Subgroups conjugate in a space group š¯’¢ are subgroups that are equivalent by symmetry operations of š¯’¢. This can be orientational or translational conjugation. The normalizer of a subgroup ā„‹ in the group š¯’¢ is the set of elements of š¯’¢ which map ā„‹ onto itself. The Euclidean normalizer of š¯’¢ is the normalizer of š¯’¢ in the Euclidean group. Euclidean normalizers of all space groups are listed in International Tables for Crystallography. Subgroups on a par are subgroups of š¯’¢ which are conjugate in the Euclidean normalizer of š¯’¢ or Z, š¯’¢ ā‰¤ Z > ā„‹; they have the same space-group type, the same cell dimensions and belong to different conjugacy classes. Different sets of coordinates for the very same crystal structure can be interconverted with the aid of the Euclidean normalizer of its space group. Chiral crystal structures can adopt only one out of 65 Sohncke space group types. The Euclidean normalizer can help to discover wrongly assigned space groups. Isotypic crystal structures have the same space group and the same distribution of their atoms.
Keywords: Ā conjugate subgroup, normalizer, Euclidean normalizer, subgroups on a par, conjugacy classes, chiral crystal structure, chiral space group, crystal structures, Sohncke space group, isotypic crystal structures
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