# Definition:Cyclic Group

## Contents

## Definition

### 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 $g \in G$:

- $G = \gen g$

## Generator

Let $a \in G$ be an element of $g$ such that $\gen a = G$.

Then $a$ is **a generator of $G$**.

## Notation

A **cyclic group** with $n$ elements is often denoted $C_n$.

Some sources use the notation $\sqbrk g$ or $\gen g$ to denote the **cyclic group** generated by $g$.

From Integers Modulo m under Addition form Cyclic Group, $\struct {\Z_m, +_m}$ is a **cyclic group**.

Thus $\struct {\Z_m, +_m}$ often taken as the archetypal example of a **cyclic group**, and the notation $\Z_m$ is used.

This is justified as, from Cyclic Groups of Same Order are Isomorphic, $\Z_m$ is isomorphic to $C_m$.

In certain contexts $\Z_m$ is particularly useful, as it allows results about **cyclic groups** to be demonstrated using number theoretical techniques.

## Examples

### Subgroup of $\struct {\R_{\ne 0}, \times}$ Generated by $2$

Consider the multiplicative group of real numbers $\struct {\R_{\ne 0}, \times}$.

Consider the subgroup $\gen 2$ of $\struct {\R_{\ne 0}, \times}$ generated by $2$.

Then $\gen 2$ is an infinite cyclic group.

### Subgroup of $\struct {\C_{\ne 0}, \times}$ Generated by $i$

Consider the multiplicative group of real numbers $\struct {\C_{\ne 0}, \times}$.

Consider the subgroup $\gen i$ of $\struct {\C_{\ne 0}, \times}$ generated by $i$.

Then $\gen i$ is an (finite) cyclic group of order $4$.

## Group Presentation

The presentation of a finite cyclic group of order $n$ is:

- $C_n = \gen {a: a^n = e}$

## Also see

- Group Generated by Singleton
- List of Elements in Finite Cyclic Group
- Order of Cyclic Group equals Order of Generator

- Results about
**cyclic groups**can be found here.

## Sources

- 1974: Thomas W. Hungerford:
*Algebra*... (previous) ... (next): $\S 1.2$ - 2009: Joseph A. Gallian:
*Contemporary Abstract Algebra*(7th ed.): Chapter $\text{IV}$