# Order (group theory)

### From Online Dictionary of Crystallography

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- | <font color="blue">Ordre</font> (''Fr'') | + | <font color="blue">Ordre</font> (''Fr''); <font color="red">Ordnung</font> (''Ge''); <font color="green">Orden</font> (''Sp''); <font color="black">Ordine</font> (''It''); <font color="purple">位数</font> (''Ja''); <font color="orange">نظام</font> (''Ar''). |

## Revision as of 09:17, 14 September 2017

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

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 has order 6 and a translation has infinite order. An element of order 2 is its own inverse and is called an involution.