Difference between revisions of "Order (group theory)"
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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 gn 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.