Translation normalizers of Euclidean groups. I. Elementary theory

Abstract

Translation normalizers of Euclidean groups are introduced as normalizers with respect to the translation subgroup of the full Euclidean group. It is shown that the concept can be easily generalized to translation normalizers of affine groups; the results for normalizers of Euclidean groups can be at once adopted for those affine groups which have orthogonalizable point groups. Some of elementary and basic properties of translation normalizers are proved. The translation normalizer is identical for all groups of the generalized arithmetic class—a concept borrowed from mathematical crystallography, because such a generalization is quite natural. It is further shown, how to use translation normalizers in consideration of sets of conjugate subgroups or supergroups and hence also of normal subgroups of Euclidean groups. For that purpose is derived the space shift diagram and normality criterion. The exposition is closed by a brief discussion of the connection of translation normalizers with location properties of Euclidean motion groups.