Q-reducible two-dimensional space groups and layer groups

Abstract

The theory of Q-reducible space groups is applied to determine those line groups in two dimensions which are isomorphic to factor groups of Q-reducible two-dimensional space groups. The concept of Q reducibility is extended to subperiodic groups, and the strip groups isomorphic to factor groups of the subperiodic layer groups are determined. The action of the point group of a space group on the underlying vector space is Q-reducible if the vector space splits into invariant subspaces such that the point group operators can be expressed by rational matrices on each subspace. It has been shown by Kopsky (1986) that if the point group is Q-reducible then (i) the point group is a subdirect product of point groups of lower dimension, (ii) the space group is a subdirect product of space groups of lower dimension and (iii) the space group has complemen- tary normal translational subgroups of lower dimension and corresponding factor groups which are isomorphic to subperiodic groups. Q reducibility gives insight into the structure of space groups with Q-reducible point groups and can be used for the derivation of higher-dimensional space or subperiodic groups as well as for the introduction of a hierarchy of these groups (Jarrat 1980). The occurrence of subperiodic groups as factor groups is also of importance in the consideration of lattices of subgroups of space groups and can be used to develop representation theory of space groups by ascent from lower to higher dimensions (Fuksa and Kopsky 1987). We determine the subperiodic groups isomorphic to factor groups of Q-reducible two-dimensional space groups taken with respect to the normal subgroups which span each of the invariant subspaces of the underlying vector space: nine of the seventeen two-dimensional space groups are Q-reducible. These are the oblique and rectangular two-dimensional space groups numbered from one to nine in the International Tables for Crystallography (Hahn 1983). The rotational part of the symbols for these groups is given in a ZXY coordinate system where XY are the coordinates of the two- dimensional space. We shall also use the notation G2 (Bohm and Dornberger-Schiff 1967) as a notation for an arbitrary two-dimensional space group. Factor groups G2/TG,, where TGr is a one-dimensional subgroup spanning an invariant subspace, are isomorphic to line groups G2, in two dimensions. The seven groups G,, are listed along with their generators in table 1. The rotational part of the symbols of these groups are given in a ZXY coordinate system where the unique plane is the XY plane and the unique axis the X axis.