Editing Order (group theory) (section)

{{Short description|Cardinality of a mathematical group, or of the subgroup generated by an element}}
{{About|order in group theory|other uses in mathematics|Order (mathematics)|other uses|Order (disambiguation)}}
{{For|groups with an ordering relation|partially ordered group|totally ordered group}}
{{Refimprove|date=May 2011}}
{{Group theory sidebar |Finite}}
[[File:Powers of rotation, shear, and their compositions.svg|thumb|270px|Examples of [[Linear map|transformations]] with different orders: 90° rotation with order 4, [[shear mapping|shearing]] with infinite order, and their [[Function composition|compositions]] with order 3.]]
In [[mathematics]], the '''order''' of a [[finite group]] is the number of its elements. If a [[group (mathematics)|group]] is not finite, one says that its order is ''infinite''. The ''order'' of an element of a group (also called '''period length''' or '''period''') is the order of the subgroup generated by the element. If the group operation is denoted as a [[multiplicative group|multiplication]], the order of an element {{mvar|a}} of a group, is thus the smallest [[positive integer]] {{math|''m''}} such that {{math|1=''a''<sup>''m''</sup> = ''e''}}, where {{math|''e''}} denotes the [[identity element]] of the group, and {{math|''a''<sup>''m''</sup>}} denotes the product of {{math|''m''}} copies of {{math|''a''}}. If no such {{math|''m''}} exists, the order of {{math|''a''}} is infinite. 

The order of a group {{mvar|G}} is denoted by {{math|ord(''G'')}} or {{math|{{abs|''G''}}}}, and the order of an element {{math|''a''}} is denoted by {{math|ord(''a'')}} or {{math|{{abs|''a''}}}}, instead of <math>\operatorname{ord}(\langle a\rangle),</math> where the brackets denote the generated group.

[[Lagrange's theorem (group theory)|Lagrange's theorem]] states that for any subgroup {{math|''H''}} of a finite group {{math|''G''}}, the order of the subgroup divides the order of the group; that is, {{math|{{abs|''H''}}}} is a [[divisor]] of {{math|{{abs|''G''}}}}. In particular, the order {{math|{{abs|''a''}}}} of any element is a divisor of {{math|{{abs|''G''|}}}}.