Definition:Infinite Cyclic Group/Definition 2

Definition
An infinite cyclic group is a cyclic group $G$ such that:
 * $\forall a \in G: \forall m, n \in \Z: m \ne n \implies a^m \ne a^n$

That is, such that all the powers of $a$ are distinct.

Also see

 * Equivalence of Definitions of Infinite Cyclic Group