# 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 .

The order of an element $g\in G$ is the smallest positive integer $n$ such that $g^{n}=e$, where $e$ is the identity element; 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)