Subperiodic groups as factor groups of reducible space groups

Abstract

Reducible space groups are introduced as those for which the point groups G are Q-reducible. The splitting of the rational space V(T, Q), spanned by the translation subgroup T of the reducible space group, into two G-invariant components is considered. First, it is shown that cases of orthogonal and inclined reductions have to be distinguished. Further, reductions which lead to Z decomposition are distinguished from those which lead to Z reduction. The central point of the paper is the 'factorization theorem' which asserts that factor groups of reducible space groups by their partial G-invariant translation subgroups have the structure of subperiodic groups. The homomorphisms which map the space group onto respective subperiodic groups are analogous to homomorphisms, which map space groups onto respective site-point groups. In analogy with point groups, subperiodic groups are introduced which do not act on the Euclidean space but on a Cartesian product of Euclidean space spanned by their translation subgroup with the vector space spanned by missing translations; it is suggested that these groups are called the contracted subperiodic groups and a formalism is developed in which these groups are geometrically natural representatives of factor groups of reducible space groups.