\begin{definition}[Definition:Order of Group Element/Definition 3]
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:
:$\forall i, j \in \Z: 0 \le i < j < k \implies x^i \ne x^j$
\end{definition}