The application of Hermann's group M in group - subgroup relations between space groups

This paper is devoted to the study of the group-subgroup relations U < G between space groups. A procedure has been developed for the derivation of all subgroups U-j < G which are obtained from U by a transformation with a translation (T-equivalent subgroups). All T-equivalent supergroups G(k)>U can be determined in the same way from one supergroup G > U. The decisive group in this procedure is the translation part of the (Euclidean) normalizer of Hermann's group M. The group M is the uniquely determined group U less than or equal toM less than or equal toG with the translations of G and the point group of U. The method is particularly useful in the search for supergroups of space groups and is based on several lemmata which are formulated and proven in this paper. The results suggest under special conditions the possibility of a transition with 'region' formation in some analogy to the well known domain formation. This transition could occur from high symmetry to low symmetry or from low symmetry to high symmetry or even both ways.