Choice of Origin, Location of Group and Uniqueness of Hermann-MauguinSymbols

[摘要] It is shown that widely accepted opinion which says that Euclidean groups cannot be located in space because they are abstract groups, is incorrect. Euclidean groups are quite certain groups of operators in space and as the symmetry operations have its location di ering by fractions of the translation normal-izer of the group the group itself has also a certain location. We show, however that derivation of space groups with the use of factor systems indeed leads to groups which are not located while derivation with the use of systems of non-primitive translations leads to a set of groups di ering in location by a fraction of their Euclidean normalizer. Some examples of possible use are given.

[关键词] Symmetry;Crystallographers;Translation normalizer;Euclidean space.