Point groups in crystallography

Two dual spaces are extensively used in crystallography: the point space E(n), hosting the crystal pattern; and the vector space V(n), where face normals and reciprocal-lattice vectors are defined. The term point group is used in crystallography to indicate four different types of groups in these two spaces. 1) Morphological point groups in V(n); they can be obtained by determining subgroups of maximal holohedries (holohedries not in group-subgroup relation): this gives 21 and 1:36 point groups in V(2) and V(3), respectively, which are classified into 10 and 32 point-group types (on the basis of which geometric crystal classes are defined) falling into 9 and 18 abstract isomorphism classes. 2) Symmetry groups of atomic groups and coordination polyhedra in E(n); they coincide with molecular point groups, which are infinite in number because the symmetry operations forming these groups are not subject to the crystallographic restriction. 3) Site-symmetry groups in E(n); they are finite groups but infinite in number due to conjugation by the translation subgroups of the space groups. They are classified in geometric crystal classes exactly like point groups in V(n). A liner classification of site-symmetry groups into species is however introduced that takes into account their orientation in space: species of site-symmetry groups in E(n) uniquely correspond to point groups in V(n). 4) Groups of matrices representing the linear parts of space group operations in E(n); they are isomorphic to the point groups in V(n) and are also isomorphic to the factor groups G/T, where G is a space group and T its translation subgroup.