\begin{definition}[Definition:Infinite Cyclic Group/Definition 2]
An '''infinite cyclic group''' is a cyclic group $G$ such that:
:$\forall a \in G, a \ne e: \forall m, n \in \Z: m \ne n \implies a^m \ne a^n$
where $e$ is the identity element of $G$.
That is, such that all the powers of $a$ are distinct.
\end{definition}