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:


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