Lattices of normal subgroups of space and subperiodic groups

Abstract

This is an introductory paper to a series in which we shall describe lattices of normal subgroups of discrete Euclidean motion groups up to three dimensions. Elements of lattice theory, especially basic properties of modular lattices are recalled. After this we outline the plan of investigation, the centre of which lies in the determination of lattices of normal translation subgroups. A suitable approach consists in considering the translation groups as operator groups and normal subgroups as theirG-admissible subgroups, whereG is the point group of the space group. The reducibility ofG over the fieldsC, R, Q and over the ringZ plays a great role in this consideration. In particular, we show that a space group has a finite number of normal subgroups with nontrivial point group just if its own point group is Q-irreducible. Lattices of subperiodic groups can be derived from lattices of underlying space groups. Referring to another recent result we show that the lattices of subperiodic groups appear as sublattices in lattices of space groups withQ-reducible point groups. This can be used to reveal at least partially the structure of lattices of space groups withQ-reducible point groups. The paper is closed by a brief discussion of the connection of lattices of normal subgroups of space groups with their representations.