# 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.

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 known as

A **cyclic group** is also known as a **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:

- $G = \set {\ldots, a^{-3}, a^{-2}, a^{-1}, e, a, a^2, a^3, \ldots}$