The implications of normalizers on group–subgroup relations between space groups

A hierarchy of classifications for subgroups of space groups by means of Euclidean and affine normalizers is introduced. The different levels of this classification scheme are illustrated in detail with examples and its usefulness for various problems is demonstrated. The Euclidean (or affine) normalizers of a space group G and of one of its subgroups U may either coincide [N(G)=N(U)], or form a group-subgroup pair [N(G) ⊃ N(U) or N(G) ⊂ N(U)], or share only a common subgroup [N(G) ⊇N(U) and N(G) ¢ N(U)]. The different implications of these cases on the equivalence classes of subgroups (or supergroups) are discussed. A procedure is given to calculate the normalizers. The same concept may be applied to number of equivalent subgroups or supergroups, other crystallographic groups without problems.