# Order (group theory)

### From Online Dictionary of Crystallography

##### Revision as of 13:19, 16 November 2017 by BrianMcMahon (talk | contribs) (moved Order to Order (group theory): Disambiguation)

Revision as of 13:19, 16 November 2017 by BrianMcMahon (talk | contribs) (moved Order to Order (group theory): Disambiguation)

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

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.