# Left Cosets and Right Cosets: Definition, Examples, Properties, Theorems

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.

## 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.

1. For h ∈ H, the corresponding left (or right) coset is H, that is, hH=H=Hh.
2. H itself a left coset (or a right coset).
3. For h $\not \in$ H, the coset hH and H are distinct.
4. Two cosets are either identical or disjoint.
5. Any two left (or right) cosets have the same cardinality.
6. For a fixed subgroup H of G, the left cosets form a partition of G.
7. 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

## 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-1b ∈ H.

Proof:

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

|aH|=|bH|

Proof:

## 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

1. 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.
2. Two double cosets are either identical or disjoint. In other words, we have either HxK = HyK or HxK HyK = ∅ for x, y ∈ G.
3. For fixed subgroups H and K of G, the set of all double cosets form a partition of G.