Equivalence of Definitions of Cyclic Group
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Theorem
The following definitions of the concept of Cyclic Group are equivalent:
Definition 1
The group $G$ is cyclic if and only if every element of $G$ can be expressed as the power of one element of $G$:
- $\exists g \in G: \forall h \in G: h = g^n$
for some $n \in \Z$.
Definition 2
The group $G$ is cyclic if and only if it is generated by one element $g \in G$:
- $G = \gen g$
Proof
$(1)$ implies $(2)$
Let $G$ be a Cyclic Group by definition 1.
Then by definition:
From $(1)$, it follows from Group Axiom $\text G 0$: Closure that:
- $\gen g \subseteq G$
From $(2)$ it follows that:
- $G \subseteq \gen g$
Thus:
- $G = \gen g$
and $G$ is a Cyclic Group by definition 2.
$\Box$
$(2)$ implies $(1)$
Let $G$ be a Cyclic Group by definition 2.
Then by definition:
- $G = \gen g$
Thus as $g^1 \in \gen g$
- $g \in G$
and by definition of generator:
- $\forall h \in \gen g: h = g^n$
for some $n \in \Z$.
Thus $G$ is a Cyclic Group by definition 1.
$\blacksquare$
Sources
- 1978: Thomas A. Whitelaw: An Introduction to Abstract Algebra ... (previous) ... (next): $\S 39.1$ Cyclic groups