Space group

Groupe d'espace (Fr). Raumgruppe (Ge). Gruppo spaziale (It). 空間群 (Ja).

The symmetry group of a three-dimensional crystal pattern is called its space group. In E², the symmetry group of a two-dimensional crystal pattern is called its plane group. In E¹, the symmetry group of a one-dimensional crystal pattern is called its line group.

To each crystal pattern belongs an infinite set of translations T, which are symmetry operations of that pattern. The set of all T forms a group known as the translation subgroup T of the space group G of the crystal pattern. T is an Abelian group and a normal subgroup of the space group. The factor group G/T of a space group G and its translation subgroup is isomorphic to the point group P of G.

See also

• Fixed-point-free space groups
• Symmorphic space groups
• Chapter 1.3 of International Tables for Crystallography, Volume A, 6th edition