---
tag: 05FD317V2W
kind: definition
name: Left coset
msc: [20A05]
defines: 03DEH43F4H
uses: [0N0PXCTYA7]
sources:
  - url: https://en.wikipedia.org/wiki/Coset
    locator: Definition
  - url: https://en.wikipedia.org/wiki/Coset
    locator: Properties
---

## Definition

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

## Remarks

Two left cosets $aH$ and $bH$ are equal when $a^{-1}b \in H$, and they are disjoint otherwise.
Thus the left cosets of $H$ form a partition of $G$, and $H = eH$ is one of them.
In $(\mathbb{Z}, +)$, the left cosets of $3\mathbb{Z}$ are $3\mathbb{Z}$, $1 + 3\mathbb{Z}$ and $2 + 3\mathbb{Z}$.

## Explanations

### L1 human

A left coset is a copy of the subgroup that you move by multiplying with a fixed element on the left.
Different elements can give the same coset.
For the even integers inside the integers, there are two cosets: the even integers and the odd integers.