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.
Table of Contents
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 Groups Kernel 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-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:
Define a mapping φ: H → aH by φ(h)=ah.
Take h1, h2 ∈ H.
Now, φ(h1) = φ(h2) ⇔ ah1 = ah2 ⇔ h1 = h2.
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.
