# Coset

From Groupprops

This article defines a property of subsets of groups

View other properties of subsets of groups|View properties of subsets of abelian groups|View subgroup properties

## Contents

## Definition

### Symbol-free definition

A subset of a group is said to be a **coset** if it satisfies the following equivalent conditions:

- It occurs as a left coset of some subgroup, or equivalently, its left quotient is a subgroup
- It occurs as a right coset of some subgroup, or equivalently, its right quotient is a subgroup
- The translates of the subset under left multiplication by elements of the group are pairwise disjoint and form a partition of the whole group
- The translates of the subset under right multiplication by elements of the group are pairwise disjoint, and form a partition of the whole group.

### Equivalence of definitions

`Further information: equivalence of definitions of coset`