---
tag: 060Z37VVB6
kind: definition
name: Right coset
msc: [20A05]
defines: 0NE1DK7H4T
uses: [0N0PXCTYA7]
sources:
  - url: https://en.wikipedia.org/wiki/Coset
    locator: Definition
---

## Definition

Let $G$ be a [[group|G5S1M9]], let $H$ be a [[subgroup|0X1JXT47GA]] of $G$, and let $g \in G$.
The right coset of $H$ with representative $g$ is the set $Hg = \{hg : h \in H\}$.

## Remarks

Two right cosets $Ha$ and $Hb$ are equal when $ab^{-1} \in H$, and they are disjoint otherwise.
In an [[abelian group|082GHNPJ53]], each right coset $Hg$ equals the [[left coset|03DEH43F4H]] $gH$.
In general $gH$ and $Hg$ can differ, but the map $gH \mapsto Hg^{-1}$ is a bijection from the left cosets to the right cosets.

## Explanations

### L1 human

A right coset is like a left coset, but you multiply the subgroup by the fixed element on the right.
In a group where the order of products matters, left and right cosets can be different sets.
The number of left cosets and the number of right cosets are always the same.