Definition:Order of Group Element/Definition 3

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 largest $k \in \Z_{\gt 0}$ such that:
 * $\forall i, j \in \Z: 0 \le i \lt j \lt k \implies x^i \ne x^j$

Also see

 * Equivalence of Definitions of Order of Group Element