# Definition:Cyclic Group/Definition 2

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## Contents

## Definition

The group $G$ is **cyclic** if and only if it is generated by one element $g \in G$:

- $G = \gen g$

## 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.

## Also see

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

## Sources

- 1965: J.A. Green:
*Sets and Groups*... (previous) ... (next): $\S 5.4$. Cyclic groups - 1965: Seth Warner:
*Modern Algebra*... (previous) ... (next): $\S 25$ - 1967: John D. Dixon:
*Problems in Group Theory*... (previous) ... (next): Introduction: Notation - 1978: Thomas A. Whitelaw:
*An Introduction to Abstract Algebra*... (previous) ... (next): $\S 39$. Cyclic groups - 1982: P.M. Cohn:
*Algebra Volume 1*(2nd ed.) ... (previous) ... (next): $\S 3.4$: Cyclic groups - 1996: John F. Humphreys:
*A Course in Group Theory*... (previous) ... (next): Chapter $4$: Subgroups: Definition $4.7$: Notation - 2008: David Nelson:
*The Penguin Dictionary of Mathematics*(4th ed.) ... (previous) ... (next): Entry:**cyclic group**

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