Order (group theory)

From Online Dictionary of Crystallography

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<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'').
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<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 gn is the identity element of G. If no such integer exists, g is of infinite order. For example, the rotoinversion \bar 3 has order 6 and a translation has infinite order. An element of order 2 is its own inverse and is called an involution.