Coset

In mathematics, specifically group theory, given an element g of a group G and a subgroup H of G,

gH = {gh : h an element of H} is the left coset of H in G with respect to g, and
Hg = {hg : h an element of H} is the right coset of H in G with respect to g.

If the group operation is written additively, the notation used changes to g + H or H + g.

Cosets are a basic tool in the study of groups; for example they play a central role in Lagrange's theorem that states that for any finite group G, the number of elements of every subgroup H of G divides the number of elements of G.

The element g belongs to the coset gH. If x belongs to gH then xH=gH. Thus every element of G belongs to exactly one left coset of the subgroup H. Elements g and x belong to the same left coset of H if and only if g-1x belongs to H. Similar statements apply to right cosets.

If G is an abelian group, then gH = Hg for every subgroup H of G and every element g of G. In general, given an element g and a subgroup H of a group G, the right coset of H with respect to g is also the left coset of the conjugate subgroup g−1Hg with respect to g, that is, Hg = g(g−1Hg). The number of left cosets of H in G is equal to the number of right cosets of H in G. The common value is called the index of H in G.

A subgroup N of a group G is a normal subgroup of G if and only if for all elements g of G the corresponding left and right coset are equal, that is, gN = Ng. Furthermore, the cosets of N in G form a group called the quotient group or factor group.

Examples

C2

Let G = ({−1,1}, ×) be the group formed by {−1,1} under multiplication, which is isomorphic to C2, and H the trivial subgroup ({1}, ×). Then {−1} = (−1)H = H(−1) and {1} = 1H = H1 are the only cosets of H in G. Because its left and right cosets with respect to any element of G coincide, H is a normal subgroup of G.

Integers

Let G be the additive group of the integers, Z = ({..., −2, −1, 0, 1, 2, ...}, +) and H the subgroup (mZ, +) = ({..., −2m, −m, 0, m, 2m, ...}, +) where m is a positive integer. Then the cosets of H in G are the m sets mZ, mZ + 1, ..., mZ + (m − 1), where mZ + a = {..., −2m+a, −m+a, a, m+a, 2m+a, ...}. There are no more than m cosets, because mZ + m = m(Z + 1) = mZ. The coset (mZ + a, +) is the congruence class of a modulo m.[1]

Vectors

Another example of a coset comes from the theory of vector spaces. The elements (vectors) of a vector space form an abelian group under vector addition. It is not hard to show that subspaces of a vector space are subgroups of this group. For a vector space V, a subspace W, and a fixed vector a in V, the sets

${\displaystyle \{x\in V\colon x=a+n,n\in W\}}$

are called affine subspaces, and are cosets (both left and right, since the group is abelian). In terms of geometric vectors, these affine subspaces are all the "lines" or "planes" parallel to the subspace, which is a line or plane going through the origin.

Double cosets

Given two subgroups, H and K of a group G, the double cosets of H and K in G are the sets of the form HgK = {hgk : h an element of H , k an element of K }. These are the left cosets of K and right cosets of H when H=1 and K=1 respectively.[2]

Notation

Let G be a group with subgroups H and K.

1. ${\displaystyle G/H}$ denotes the set of left cosets ${\displaystyle \{gH:g\in G\}}$ of H in G.
2. ${\displaystyle H\backslash G}$ denotes the set of right cosets ${\displaystyle \{Hg:g\in G\}}$ of H in G.
3. ${\displaystyle K\backslash G/H}$ denotes the set of double cosets ${\displaystyle \{KgH:g\in G\}}$ of H and K in G.

General properties

The identity is in precisely one left or right coset, namely H itself. Thus H is both a left and right coset of itself.

A coset representative is a representative in the equivalence class sense. A set of representatives of all the cosets is called a transversal. There are other types of equivalence relations in a group, such as conjugacy, that form different classes which do not have the properties discussed here.

Index of a subgroup

Every left or right coset of H has the same number of elements (or cardinality in the case of an infinite H) as H itself. Furthermore, the number of left cosets is equal to the number of right cosets and is known as the index of H in G, written as [G : H]. Lagrange's theorem allows us to compute the index in the case where G and H are finite:

${\displaystyle |G|=[G:H]|H|}$.

This equation also holds in the case where the groups are infinite, although the meaning may be less clear.

Cosets and normality

If H is not normal in G, then its left cosets are different from its right cosets. That is, there is an a in G such that no element b satisfies aH = Hb. This means that the partition of G into the left cosets of H is a different partition than the partition of G into right cosets of H. (Some cosets may coincide. For example, if a is in the center of G, then aH = Ha.)

On the other hand, the subgroup N is normal if and only if gN = Ng for all g in G. In this case, the set of all cosets form a group called the quotient group G / N with the operation ∗ defined by (aN) ∗ (bN) = abN. Since every right coset is a left coset, there is no need to distinguish "left cosets" from "right cosets".