order (of a group)

The order of a group G is the number of elements of G, denoted |G|; if |G| is finite, then G is said to be a finite groupMathworldPlanetmath.

The order of an element gG is the smallest positive integer n such that gn=e, where e is the identity elementMathworldPlanetmath; if there is no such n, then g is said to be of infinite order. By Lagrange’s theorem, the order of any element in a finite group divides the order of the group.

Title order (of a group)
Canonical name OrderofAGroup
Date of creation 2013-03-22 12:36:47
Last modified on 2013-03-22 12:36:47
Owner mathcam (2727)
Last modified by mathcam (2727)
Numerical id 9
Author mathcam (2727)
Entry type Definition
Classification msc 20A05
Synonym order
Related topic Group
Related topic Cardinality
Related topic OrdersOfElementsInIntegralDomain
Related topic OrderRing
Related topic IdealOfElementsWithFiniteOrder
Defines finite group
Defines infinite order
Defines order (of a group element)