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