Complete normalizer for a direct product of three-dimensional rotation groups

A normalizer of the symmetry group defined on a three-dimensional sphere S-3 of rotation is considered in the four-dimensional Euclidean space E-4. The sphere S-3 is treated as the first approximation of the three-dimensional crystallographic space. The analysis of the normalizer N of the direct product G = G(1) X G(2) of space crystallographic rotation groups G(1) and G(2) is reduced to the study of transformations characterized by the positive determinants of the subgroups N+(G(1)) and N+(G(2)) These subgroups correspond to the Euclidean normalizers N = N+(G(1)) x N+(G(2)) of the components of the direct product. We derived a table including the groups of automorphisms induced by the transformations corresponding to the normalizers under study. Analyzing the general operation of multiplication of three-dimensional rotations in E-4, We refined the distribution of the supersymmetry operators of the three-dimensional sphere of rotations, S-3, for the symmetry groups considered earlier. (C) 2000 MAIK "Nauka/Interperiodica".