---
tag: BM4VXG92DT
kind: example
name: The cosets of 3Z in the integers
msc: [20A05]
instance_of: [03DEH43F4H]
uses: [0C4JS9RF5M, 05FD317V2W, 03VFZ2WETT]
sources:
  - url: https://en.wikipedia.org/wiki/Coset
    locator: Integers
---

## Example

Let $G = (\mathbb{Z}, +)$ be the [[integers|0C4JS9RF5M]] with addition, and let $H = 3\mathbb{Z}$ be the multiples of $3$.
The [[cosets|05FD317V2W]] of $H$ are $3\mathbb{Z}$, $1 + 3\mathbb{Z}$ and $2 + 3\mathbb{Z}$.
Each integer is in exactly one of them, given by its remainder after division by $3$.
The group is abelian, so each left coset is also a right coset.

## Remarks

The three cosets are the elements of the group [[$\mathbb{Z}/3\mathbb{Z}$|03VFZ2WETT]].