Definition:Infinite Cyclic Group

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Definition

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.


Group Presentation

The presentation of an infinite cyclic group is:

$G = \gen a$

This specifies $G$ as being generated by a single element of infinite order.


Also known as

An infinite cyclic group is also known as a free group on one generator.


From Integers under Addition form Infinite Cyclic Group, the additive group of integers $\struct {\Z, +}$ forms an infinite cyclic group.

Thus the notation $\Z$ is often used for the infinite cyclic group.

This is justified as, from Cyclic Groups of Same Order are Isomorphic, $\Z$ is isomorphic to $\gen a$.


Also see

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