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:

$(1): \quad g \in G$
$(2): \quad$ every element of $G$ is expressible as a power of $g$.

From $(1)$, it follows from Group Axion $G 1$: 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.

$\blacksquare$


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