Cosets are mainly used to decompose a group G into equal-sized disjoint subsets of G. It plays an important role to study many things in Group Theory; for example, normal group, Lagrange’s theorem on finite groups, etc. In this post, we will learn about cosets, their classification with examples, and their properties with related theorems.

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## Definition of Cosets

A coset of a subgroup H of a group (G, o) is a subset of G obtained by multiplying H with elements of G from left or right.

**For example**, take H=(Z, +) and G=(Z, +). Then 2+Z, Z+6 are cosets of H in G.

Depending upon the multiplication from left or right we can classify cosets as left cosets or right cosets as follows:

## Definition of Left Cosets

Let G be a group and H be a subgroup of G. Then for an element g** **∈ G, the left coset of H in G is defined by

gH = {gh : h ∈** **H}.

The set gH is called a left coset of H in G, and the element g is called a representative of the left coset gH.

## Definition of Right Cosets

Let H be a subgroup of a group G. Then the set defined below

Hg = {hg : h ∈** **H} where g **∈ **G,

is called a right coset of H in G. The element g is called a representative of the right coset Hg.

## Examples of Left Cosets and Right Cosets

Let G=(Z, +) and H=(2Z, +). The following are examples of the left cosets of H in G.

- 0+H = {2n : n ∈ Z} = H
- 1+H = {2n+1 : n ∈ Z}
- 2+H = {2n+2 : n ∈ Z}

Below are examples of the right cosets of H in G.

- H+0 = {2n : n ∈ Z} = H
- H+1 = {2n+1 : n ∈ Z}
- H+2 = {2n+2 : n ∈ Z}

## Properties of Cosets

Let G be a group and H be its subgroup. The following are a few properties of left cosets and right cosets.

- For h ∈ H, the corresponding left (or right) coset is H, that is, hH=H=Hh.
- H itself a left coset (or a right coset).
- For h $\not \in$ H, the coset hH and H are distinct.
- Two cosets are either identical or disjoint.
- Any two left (or right) cosets have the same cardinality.
- For a fixed subgroup H of G, the left cosets form a partition of G.
- Normal subgroups are defined using the concept of cosets. For a normal subgroup N of G, the set of all left cosets of N in G form a group, called the quotient group and it is denoted by G/N.

## Applications of Cosets

- Cosets play a crucial role in computational group theory, for example, to calculate the index of a subgroup.
- Cosets are an important tool to prove Lagrange’s theorem for finite groups.
- Vitali sets are not measurable which are constructed using cosets.
- Thistlethwaite’s algorithm is used to solve Rubik’s Cube. It is purely based on the theory of cosets

Also Read:Order of a Group: The order of a group and of its elements are discussed here with formulas. Abelian Group: Definition, Properties, Examples Cyclic Group: The definition, properties, and related theorems on cyclic groups are discussed.First Isomorphism Theorems of GroupsKernel of a Group Homomorphism |

## Theorems on Cosets

Let H be a subgroup of G. Then we have the following theorems on cosets.

Theorem 1: hH=H for h ∈ H. |

*Proof:*

Theorem 2: aH ∩ H = ∅ for a ∈ G-H. |

*Proof:*

Theorem 3: Two cosets are either identical or disjoint, that is,aH=bH or aH ∩ bH = ∅. |

*Proof:*

Theorem 4: (Equality of Cosets)aH=bH if and only if a ^{-1}b ∈ H. |

*Proof:*

Theorem 5: (Cardinality of Cosets) Any two left cosets of H in G have the same cardinality, that is,|aH|=|bH| |

*Proof:*

Define a mapping φ: H → aH by φ(h)=ah.

Take h_{1}, h_{2} ∈ H.

Now, φ(h_{1}) = φ(h_{2}) ⇔ ah_{1} = ah_{2} ⇔ h_{1} = h_{2}.

This shows that φ is one-to-one.

Let y ∈ aH. Then y=ah for some h∈ H. The element y has the pre-image h, that is, φ(h)=ah. This makes φ onto.

Thus we have a bijection between H and aH.

Similarly, there is a bijection between H and bH.

As a result, aH and bH have the same cardinality as that of H. ♣

**Remark:**

By Theorem 5, we can conclude that each left or right coset of H in G has the same cardinality as that of H. Note that H can be regarded as both left or right cosets.

## Double Cosets

Let G be a group and H, K be its two subgroups. Then form some g **∈** G, the set

HgK = {hgk : h ∈** **H, k ∈** **K}

is called a double coset of H and K in G.

### Properties of Double Cosets

- If H=1, then the double cosets are actually left cosets of K in G. Similarly, if K=1, then the double cosets are right cosets of H in G.
- Two double cosets are either identical or disjoint. In other words, we have either HxK = HyK or HxK
**∩**HyK = ∅ for x, y ∈ G. - For fixed subgroups H and K of G, the set of all double cosets form a partition of G.