Order (group theory)
From Online Dictionary of Crystallography
Revision as of 10:30, 16 May 2017 by BrianMcMahon (Style edits to align with printed edition)
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.