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In this article, a number of the results relevant to the concept of sublattices of a basic crystallographic lattice are reviewed, emphasizing particularly previously unpublished work on the algebraic aspects. A three-dimensional geometric lattice L can be considered as an infinite Abelian group under addition. A sublattice S of L, which is also three-dimensional, is a subgroup of L such that the finite quotient group, is an Abelian group of order the index of S in L. The sublattice itself in its standard form is represented by an upper triangular matrix. The index of the sublattice is given by the determinant of this matrix. It is first noted that a sublattice described by an arbitrary basis set in L may be converted to this standard form. Next the sublattice is expressed as the intersection of a set of sublattices of individual index a power of a distinct prime, i.e. where p1, p2 etc. are prime numbers and is the Euclidean factorization of n. This decomposition is important because it corresponds to the Sylow decomposition of the corresponding quotient group It is also useful to be able to carry out two commutative binary operations on sublattices of L; these are to find their common sublattice of lowest index in L, which is their intersection and their common superlattice of highest index in L, given by where <> indicates the span of the sublattices. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]