Category:Infinite Cyclic Group

This category contains results about the infinite cyclic group.
Definitions specific to this category can be found in Definitions/Infinite Cyclic Group.

Definition 1

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

$\forall n \in \Z_{> 0}: n > 0 \implies \nexists a \in G, a \ne e: a^n = e$

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.