Definition:Cyclic Group

Definition
A group $$G$$ is cyclic if there exists $$g \in G$$ such that for every $$h \in G$$, $$h = g^n$$ for some positive integer $$n$$.

That is, if every element of $$G$$ is a power of a fixed element of that group.

We say that $$g$$ generates $$G$$ and write $$G = \left \langle {g}\right \rangle$$.

It follows directly from Results concerning Order of Element that if $$\left|{G}\right| = m$$ then $$G = \left\{{e, g, g^2, \ldots, g^{m - 1}}\right\}$$ and $$\left|{g}\right| = m$$.

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 $$g$$.

As the Additive Group of Integers Modulo m is a cyclic group, the notation $$\Z_m$$ is often used.

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

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:


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