# Equivalence of Definitions of Cyclic Group

Jump to navigation
Jump to search

## 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