Definition:Order of Group Element

Definition
Let $G$ be a group whose identity is $e_G$.

Let $x \in G$ be an element of $G$.

The order of $x$ (in $G$), denoted $\left\vert{x}\right\vert$, is the smallest $k \in \Z_{> 0}$ such that $x^k = e_G$.

That is, such that:
 * $\forall i, j \in \Z: 0 \le i < j < k \implies x^i \ne x^j$

Also known as
Some sources call this the period of the element.

Also denoted as
The order of an element $g \in G$ is sometimes seen as $o \left({g}\right)$.

Some sources render it as $\operatorname{Ord} \left({g}\right)$.

Also see

 * Equal Powers of Group Element implies Finite Order