\begin{definition}[Definition:Order of Group Element/Infinite/Definition 2]
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 powers $x, x^2, x^3, \ldots$ of $x$ are all distinct:
:$\order x = \infty$
\end{definition}