# Space group

### From Online Dictionary of Crystallography

Groupe d'espace (*Fr*); Raumgruppe (*Ge*); Gruppo spaziale (*It*); 空間群 (*Ja*); صنف أو مجموعة الفضاء (*Ar*).

The symmetry group of a three-dimensional crystal pattern is called its **space group**. In *E*^{2}, the symmetry group of a two-dimensional crystal pattern is called its **plane group**. In *E*^{1}, 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