Definition:Infinite Cyclic Group/Definition 2
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Definition
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.
Also see
- Results about the infinite cyclic group can be found here.
Sources
- 1965: J.A. Green: Sets and Groups ... (previous) ... (next): $\S 5.4$. Cyclic groups
- 1971: Allan Clark: Elements of Abstract Algebra ... (previous) ... (next): Chapter $2$: Subgroups and Cosets: $\S 43$