---
tag: 0G0B3DRG65
kind: example
name: Cyclic group of order n
aliases: [integers-modulo-n-under-addition]
msc: [20K01]
instance_of: [0KVGH0J5B3]
uses: [0B85Q41RP6, 0SEX9F9J5Z]
sources:
  - url: https://en.wikipedia.org/wiki/Cyclic_group
    locator: Integer and modular addition
---

## Example

Let $n$ be a positive integer.
The integers modulo $n$, $\mathbb{Z}/n\mathbb{Z} = \{0, 1, \ldots, n - 1\}$, with addition modulo $n$, form a [[cyclic group|0KVGH0J5B3]] of [[order|0B85S2V2DR]] $n$.
The residue $1$ generates it, because each residue $k$ is the sum of $k$ copies of $1$.
Each cyclic group of order $n$ is isomorphic to $\mathbb{Z}/n\mathbb{Z}$, and this group is often written $C_n$.

## Remarks

A residue $k$ generates $\mathbb{Z}/n\mathbb{Z}$ exactly when $\gcd(k, n) = 1$.
The [[group of $n$th roots of unity|0WX11EA3F3]] is a cyclic group of order $n$ in multiplicative notation.
The rotations of a regular $n$-gon also form a cyclic group of order $n$.

## Explanations

### L1 human

Think of a clock with $n$ hours, where you add hours and wrap around after $n$.
This clock arithmetic gives the cyclic group of order $n$.
Every cyclic group of order $n$ looks the same as this clock, up to the names of the elements.