View source for Order (group theory)

<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'').


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 [[involution]].




[[Category:Fundamental crystallography]]