\begin{definition}[Definition:Order of Group Element/Infinite/Definition 3]
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}} the group $\gen x$ generated by $x$ is of infinite order.
:$\order x = \infty \iff \order {\gen x} = \infty$
\end{definition}