Order of a group element

Given an element \(g\) of group \((G, +)\) (which henceforth we abbreviate simply as \(G\)), the order of \(g\) is the number of times we must add \(g\) to itself to obtain the identity element \(e\).

knows-requisite(Order of a group): Equivalently, it is the order of the group \(\langle g \rangle\) generated by \(g\): that is, the order of \(\{ e, g, g^2, \dots, g^{-1}, g^{-2}, \dots \}\) under the inherited group operation \(+\).

Conventionally, the identity element itself has order \(1\).


knows-requisite(Symmetric group): In the symmetric group \(S_5\), the order of an element is the least common multiple of its cycle type.
knows-requisite(Cyclic group): In the cyclic group \(C_6\), the order of the generator is \(6\). If we view \(C_6\) as being the integers modulo \(6\) under addition, then the element \(0\) has order \(1\); the elements \(1\) and \(5\) have order \(6\); the elements \(2\) and \(4\) have order \(3\); and the element \(3\) has order \(2\).

In the group \(\mathbb{Z}\) of integers under addition, every element except \(0\) has infinite order.\(0\) itself has order \(1\), being the identity.


  • Group

    The algebraic structure that captures symmetry, relationships between transformations, and part of what multiplication and addition have in common.