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 <tt><span style="text-decoration: overline">3</span></tt> has order 6 and a translation has infinite order. An element of order 2 is called an '''involution'''. | + | 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 <tt><span style="text-decoration: overline">3</span></tt> 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 13:08, 8 December 2014
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 3 has order 6 and a translation has infinite order. An element of order 2 is its own inverse and is called an involution.