# Order (group theory)

### From Online Dictionary of Crystallography

##### Revision as of 10:30, 16 May 2017 by BrianMcMahon (talk | contribs) (Style edits to align with printed edition)

Revision as of 10:30, 16 May 2017 by BrianMcMahon (talk | contribs) (Style edits to align with printed edition)

Ordre (*Fr*). Ordnung (*Ge*). Orden (*Sp*). Ordine (*It*). 位数 (*Ja*).

If *G* is a group consisting of a finite number of elements, this number of elements is the **order** of *G*. For example, the point group `m3m` has order 48.

For an element *g* of a (not necessarily finite) group *G*, the **order** of *g* is the smallest integer *n* such that *g ^{n}* is the identity element of

*G*. If no such integer exists,

*g*is of

**infinite order**. For example, the rotoinversion [math]\bar 3[/math] has order 6 and a translation has infinite order. An element of order 2 is its own inverse and is called an

**involution**.