# Definition:Cyclic Group/Definition 1

## Contents

## Definition

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 $n \in \Z$.

## 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

- 1964: Walter Ledermann:
*Introduction to the Theory of Finite Groups*(5th ed.) ... (previous) ... (next): $\S 9$: Cyclic Groups: Definition $6$ - 1965: J.A. Green:
*Sets and Groups*... (previous) ... (next): $\S 5.4$. Cyclic groups - 1966: Richard A. Dean:
*Elements of Abstract Algebra*... (previous) ... (next): $\S 1.7$ - 1967: George McCarty:
*Topology: An Introduction with Application to Topological Groups*... (previous) ... (next): $\text{II}$: A Little Number Theory - 1971: Allan Clark:
*Elements of Abstract Algebra*... (previous) ... (next): Chapter $2$: Subgroups and Cosets: $\S 43$ - 1972: A.G. Howson:
*A Handbook of Terms used in Algebra and Analysis*... (previous) ... (next): $\S 5$: Groups $\text{I}$ - 1978: John S. Rose:
*A Course on Group Theory*... (previous) ... (next): $0$: Some Conventions and some Basic Facts - 1978: Thomas A. Whitelaw:
*An Introduction to Abstract Algebra*... (previous) ... (next): $\S 39.1$ Cyclic groups - 1982: P.M. Cohn:
*Algebra Volume 1*(2nd ed.) ... (previous) ... (next): $\S 3.4$: Cyclic groups

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