# Point group

Groupe ponctuel (Fr); Punktgruppe (Ge); Gruppo punto (It); 点群 (Ja).

## 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 rotations and rotoinversions are restricted to 1, 2, 3, 4, 6 and $\bar 1$, $\bar 2$ (= m), $\bar 3$, $\bar 4$, $\bar 6$ 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 polyedra ("point point group").