Definition:Infinite Cyclic Group

Definition
An infinite cyclic group is a cyclic group $G$ such that:
 * $\forall n \in \N^*: n > 2 \implies \not \exists a \in G, a \ne e: a^n = e$

Alternatively:
 * $\forall a \in G: \forall m, n \in \Z: m \ne n \implies a^m \ne a^n$

The two definitions are equivalent.

Group Presentation
The presentation of an infinite cyclic group is:


 * $G = \left \langle {a} \right \rangle$

This specifies $G$ as being generated by a single element of infinite order.

As the Additive Group of Integers $\left({\Z, +}\right)$ forms an infinite cyclic group, the notation $\Z$ is often used for the infinite cyclic group.

This is justified as, from Cyclic Groups Same Order Isomorphic, $\Z$ is isomorphic to $\left \langle {a} \right \rangle$.

Comment
This is also known as the free group on one generator.

If $G$ is an infinite cyclic group generated by $a \in G$, then $a$ is an element of infinite order, and all the powers of $a$ are different. Thus:


 * $G = \left\{{\ldots, a^{-3}, a^{-2}, a^{-1}, e, a, a^2, a^3, \ldots}\right\}$