Unified system of Hermann-Mauguin symbols for groups of material physics. 1. Groups with decomposable lattices

The system of Hermann-Mauguin symbols for space and subperiodic Euclidean groups in two and three dimensions is extended to groups with continuous and semicontinuous translation subgroups (lattices). An interpretation of these symbols is proposed in which each symbol defines a quite specific Euclidean group with reference to a crystallographic basis, including the location of the group in space. Symbols of subperiodic (layer and rod) groups are strongly correlated with symbols of decomposable space groups on the basis of the factorization theorem. Introduction of groups with continuous and semicontinuous lattices is connected with a proposal for several new terms that describe the properties of these groups and with a proposal to amend the meaning of space groups and of crystallographic groups. Charts of plane, layer and space groups describe variants of these groups with the same reducible point group but various types of lattices. Examples of such charts are given for plane, layer and space groups to illustrate the unification principle for groups with decomposable lattices.