Definition:Infinite Cyclic Group

An infinite cyclic group is a cyclic group $$G$$ such that:

$$\forall n \in \mathbb{N}^*: n > 2 \Longrightarrow \not \exists a \in G, a \ne e: a^n = e$$

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.

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:

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