# Definition:Cyclic Group/Notation

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## Definition

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.