The role of subperiodic and lower-dimensional groups in the structure of space groups

Abstract

The author considers the consequences of Q-reducibility of a point group for the structure of a corresponding space group. First he shows that the space group is a subgroup of a direct product of two Euclidean groups of lower dimensions. He introduces the so-called separation diagram from which follows that (i) the space group is a subdirect product of space groups of lower dimensions, (ii) its point group is a subdirect product of point groups of lower-dimensional groups, (iii) the space group has two complementary normal translation subgroups of lower dimensions and the corresponding factor groups are complementary subperiodic groups. The latter conclusion enables us to classify space groups into finer classes than the point classes. The most important fact is that the space groups in higher dimensions can be constructed from Q-irreducible groups of lower dimensions unless they are themselves Q-irreducible.