# Point group

In mathematics, a **point group** is a group of geometric symmetries (isometries) leaving a point fixed.

## Contents

## Overview

Point groups can exist in a Euclidean space of any dimension. A discrete point group in 2D is sometimes called a **rosette group**, and is used to describe the symmetries of an ornament. The 3D point groups are heavily used in chemistry, especially to describe the symmetries of a molecule and of orbitals forming covalent bonds, and in this context they are also called **molecular point groups**.

There are infinitely many discrete point groups in each number of dimensions. However, the crystallographic restriction theorem demonstrates that only a finite number are compatible with translational symmetry. In 1D there are 2, in 2D 10, and in 3D 32 such groups, called **crystallographic point groups**.

## In two dimensions

**Point groups in 2D** fall into two distinct families, according to whether they consist of rotations only, or include reflections. The *cyclic groups*, C_{n} (abstract group type Z_{n}), consist of rotations by 360°/*n*, and all integer multiples. For example, a swastika has symmetry group C_{4}, consisting of rotations by 0°, 90°, 180°, and 270°. The symmetry group of a square belongs to the family of *dihedral groups*, D_{n} (abstract group type Dih_{n}), including as many reflections as rotations. The infinite rotational symmetry of the circle implies reflection symmetry as well, but formally the circle group S_{1} is distinct from Dih(S_{1}) because the latter explicitly includes the reflections.

An infinite group need not be continuous; for example, we have a group of all integer multiples of rotation by 360°/√2, which does not include rotation by 180°. Depending on its application, homogeneity up to an arbitrarily fine level of detail in a transverse direction may be considered equivalent to full homogeneity in that direction, in which case these symmetry groups can be ignored.

C_{n} and D_{n} for *n* = 1, 2, 3, 4, and 6 can be combined with translational symmetry, sometimes in more than one way. Thus these 10 groups give rise to 17 wallpaper groups.

## In three dimensions

More complex symmetries arise in 3D, see **point groups in three dimensions**.

## Generalization

In any dimension *d*, the continuous group of all possible fixed point isometries is the *orthogonal group*, denoted by O(*d*); and its continuous subgroup of all possible rotations is the *special orthogonal group*, denoted by SO(*d*). This is not Schönflies notation, but the conventional names from Lie group theory.

## See also

- Crystallography
- Crystallographic point group
- Molecular symmetry
- Wallpaper group
- Space group
- X-ray diffraction
- Bravais lattice

## External links

- Downloadable point group tutorial (Mac and Windows only)
- Molecular symmetry examples
- Web-based point group tutorial (needs Java and Flash)
- Subgroup enumeration (needs Java)