\begin{definition}[Definition:Order of Group Element/Infinite/Definition 1]
Let $G$ be a group whose identity is $e_G$.
Let $x \in G$ be an element of $G$.
$x$ '''is of infinite order''', or '''has infinite order''' {{iff}} there exists no $k \in \Z_{> 0}$ such that $x^k = e_G$:
:$\order x = \infty$
\end{definition}