Definition:Infinite Cyclic Group/Definition 1

Definition

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

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

Also see

• Equivalence of Definitions of Infinite Cyclic Group

• Results about the infinite cyclic group can be found here.

Sources

• 1966: Richard A. Dean: Elements of Abstract Algebra ... (previous) ... (next): $\S 1.7$: Example $24$
• 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): generator: 2. (of a group)
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): generator: 2. (of a group)