Space groups mathematical definitions

The ensemble of all equivalent positions for a space group is unique and may be considered the mathematical definition of the space group. It provides the basis for manipulating objects and points related by symmetry in a digital computer. Equivalent positions are another way of stating both the space group and the Bravais lattice of a crystal. 

In a further development of detail, one can take into account how the atoms of the solid are distributed spatially. The issue of symmetry in context with a fixed point in the crystal, and the symmetry of Bravais lattices, has been addressed, but in order to describe the entire crystal the effects of two new types of symmetry operation must be included. A space group determined in this way describes the spatial symmetry of the crystal. By definition, a crystallographic space group is the set of geometrical symmetry operations that take a three-dimensional periodic crystal into itself The set of operations that make up the space group must form a group in the mathematical sense and must include the primitive lattice translations as well as other symmetry operations. 

The complete group of the Hamiltonian is the combination of all these possible symmetries. This derivation is directly evident from the mathematical form of the Hamiltonian and expresses fundamental properties of molecular space and time. Yet it took 40 years, from Schrodinger to Longuet-Higgins, to obtain a clear definition of the molecular-symmetry group. Three kinds of symmetries may be identified ... 

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