# 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 $1$.

# Examples

%%%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.