# Point group

### From Online Dictionary of Crystallography

زمرة نقطية (*Ar*). Groupe ponctuel (*Fr*). Punktgruppe (*Ge*). Gruppo punto (*It*). 点群 (*Ja*). Точечная группа симметрии (*Ru*). Grupo puntual (*Sp*).

## Definition

A **point group** is a group of symmetry operations all of which leave at least one point unmoved. A *crystallographic* point group is a point group that maps a point lattice onto itself: in three dimensions the symmetry operations of these groups are restricted to 1, 2, 3, 4, 6 and [math]\bar 1[/math], [math]\bar 2[/math] (= *m*), [math]\bar 3[/math], [math]\bar 4[/math], [math]\bar 6[/math] respectively.

## Occurrence

Crystallographic point groups occur:

- in vector space, as symmetries of the external shapes of crystals (morphological symmetry), as well as symmetry of the physical properties of the crystal ('vector point group');
- in point space, as site-symmetry groups of points in lattices or in crystal structures, and as symmetries of atomic groups and coordination polyhedra ('point point group').

## Controversy on the nomenclature

The matrix representation of a symmetry operation consists of a linear part, which represents the rotation or rotoinversion component of the operation, and a vector part, which gives the shift to be added once the linear part of the operation has been applied. The vector part is divided into two components: the *intrinsic component*, which represents the screw and glide component of the operation, and the *location component*, which is non-zero when the symmetry element does not pass through the origin. The set of the linear parts of the matrices representing the symmetry operations of a space group is a representation of the point group of the crystal. On the other hand, the set of matrix-vector pairs representing the symmetry operations of a site symmetry group form a group which is isomorphic to a crystallographic point group. The vector part being in general non-zero, some authors reject the term *point group* for the site-symmetry groups. On the other hand, all the symmetry operations of a site symmetry group do leave invariant at least one point, albeit not necessarily the origin, satisfying the above definition of point group.

## See also

- Chapter 3.2.1 of
*International Tables for Crystallography, Volume A*, 6th edition