Equivalence of Definitions of Infinite Cyclic Group
Theorem
The following definitions of the concept of Infinite Cyclic Group are equivalent:
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.
Proof
$(1)$ implies $(2)$
Let $G = \gen g$ be an infinite cyclic group by definition 1.
Then by definition:
- $(1): \quad \forall n \in \Z_{> 0}: n > 0 \implies \nexists a \in G, a \ne e: a^n = e$
Aiming for a contradiction, suppose:
- $\exists a \in G: \exists m, n \in \Z, m \ne n: a^m = a^n$
Then by Equal Powers of Finite Order Element:
- $\exists k \in \Z_{> 0}: k \divides \paren {m - n}$
such that:
- $a^k = e$
As $m \ne n$, we have that $k \ge 1$.
This contradicts the condition $(1)$ for $G$ to be an infinite cyclic group.
From this contradiction we deduce that:
- $\forall a \in G: \forall m, n \in \Z: m \ne n \implies a^m \ne a^n$
Thus $G$ is an infinite cyclic group by definition 2.
$\Box$
$(2)$ implies $(1)$
Let $G = \gen g$ be an infinite cyclic group by definition 2.
Then by definition:
- $\forall a \in G: \forall m, n \in \Z: m \ne n \implies a^m \ne a^n$
Aiming for a contradiction, suppose:
- $\exists a \in G, a \ne e: \exists n \in \Z: a^n = e$
Then:
- $a^{2 n} = a^n \circ a^n = e^2 = e = a^n$
That is:
- $\exists n, 2 n \in \Z: n \ne 2 n: a^n = a^{2 n}$
and so it is not the case that:
- $\forall a \in G: \forall m, n \in \Z: m \ne n \implies a^m \ne a^n$
From this contradiction we deduce that:
- $\forall n \in \Z_{> 0}: n > 0 \implies \nexists a \in G, a \ne e: a^n = e$
Thus $G$ is an infinite cyclic group by definition 1.
$\blacksquare$
Sources
- 1966: Richard A. Dean: Elements of Abstract Algebra ... (previous) ... (next): $\S 1.7$: Theorem $8$