Definition:Cyclic Group

Definition
Let $G$ be a group.

Definition 1
The group $G$ is cyclic there exists $g \in G$ such that for every $h \in G$, $h = g^n$ for some  integer $n$, where:
 * $g^n$ is the $n$th power of $g$

Definition 2
The group $G$ is cyclic it is generated by one element $g \in G$, that is, $G = \left \langle{g}\right \rangle$

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

Some sources use the notation $\left[{g}\right]$ or $\left\langle{g}\right\rangle$ to denote the cyclic group generated by $g$.

From Integers Modulo m under Addition form Cyclic Group, $\left({\Z_m, +_m}\right)$ is a cyclic group.

Thus $\left({\Z_m, +_m}\right)$ 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.

Group Presentation
The presentation of a finite cyclic group is:


 * $C_n = \left \langle {a: a^n = e} \right \rangle$

Also see

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