---
tag: 564VVAQTNR
kind: concept
name: Coset
aliases: [co-set, coset-in-a-group, coset-representative, cosets]
sources:
  - wikipedia: en:98759@r1351557349
    locator: Coset
    note: adopted
    mention: wikipedia://en/98759
    sections: [Summary]
same_as:
  wikidata: Q751969
  wikipedia: Coset
inventory: [S8KHNG9EDR]
---

## Summary

In mathematics, specifically group theory, a [[subgroup|0X1JXT47GA]] $H$ of a group $G$ may be used to decompose the underlying [[set|0BSJ7RQ9BW]] of $G$ into [[disjoint|0W4RPA8QH3]], equal-size [[subsets|0BJR9QR08E]] called **cosets**. There are *left cosets* and *right cosets*. Cosets (both left and right) have the same number of elements ([[cardinality|0G0ASKNA6Z]]) as does $H$. Furthermore, $H$ itself is both a left coset and a right coset. The number of left cosets of $H$ in $G$ is equal to the number of right cosets of $H$ in $G$. This common value is called the [[index|0DTXBQPA2S]] of $H$ in $G$ and is usually denoted by $[G : H]$.