Definition:Order of Group Element/Definition 2

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 order of the group generated by $x$:
 * $\left\vert{x}\right\vert := \left\vert{\left\langle{x}\right\rangle}\right\vert$

Also see

 * Equivalence of Definitions of Order of Group Element