Order (group theory)

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 [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.