# Difference between revisions of "Order (group theory)"

### From Online Dictionary of Crystallography

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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 <tt>m<span style="text-decoration: overline">3</span>m</tt> has order 48. | 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 <tt>m<span style="text-decoration: overline">3</span>m</tt> 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<sup>n</sup>'' 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 | + | For an element ''g'' of a (not necessarily finite) group ''G'', the '''order''' of ''g'' is the smallest integer ''n'' such that ''g<sup>n</sup>'' 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]]. |

## Revision as of 05:29, 21 May 2017

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.