---
tag: 0B85Q41RP6
kind: definition
name: Cyclic group
msc: [20A05]
defines: 0KVGH0J5B3
uses: [0YT586TKXF]
sources:
  - url: https://en.wikipedia.org/wiki/Cyclic_group
    locator: Definition and notation
---

## Definition

A [[group|G5S1M9]] $G$ is cyclic when there is an element $g \in G$ with $G = \langle g \rangle$, where $\langle g \rangle$ is the [[cyclic subgroup generated by $g$|08ABT85339]].
Such an element $g$ is called a generator of $G$.

## Remarks

The group $(\mathbb{Z}, +)$ is cyclic, and its generators are $1$ and $-1$.
The integers modulo $n$ under addition form a cyclic group of order $n$, generated by $1$.
A cyclic group can have many generators: in the integers modulo $n$, the generators are the residues coprime to $n$.
The [[circle group|04EAFVFTF7]] is not cyclic, because it is uncountable.

## Explanations

### L1 human

A cyclic group is a group in which each element is a power of one fixed element.
The rotations of a regular polygon with $n$ sides form a cyclic group: each rotation is a power of the smallest rotation.
The word "cyclic" comes from this picture of going around a circle.