Coset

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Not to be confused with Cosette.
G is the group (Z/8Z, +), the integers mod 8 under addition. The subgroup H contains only 0 and 4, and is isomorphic to (Z/2Z, +). There are four left cosets of H: H itself, 1 + H, 2 + H, and 3 + H (written using additive notation since this is the additive group). Together they partition the entire group G into equal-size, non-overlapping sets. The index [G : H] is 4.

In mathematics, if G is a group, and H is a subgroup of G, and g is an element of G, then

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.

Only when H is normal will the set of right cosets and the set of left cosets of H coincide, which is one definition of normality of a subgroup. Although derived from a subgroup, cosets are not usually themselves subgroups of G, only subsets.

A coset is a left or right coset of some subgroup in G. Since Hg = g ( g−1Hg ), the right coset Hg (of H with respect to g) and the left coset g ( g−1Hg ) (of the conjugate subgroup g−1Hg ) are the same. Hence it is not meaningful to speak of a coset as being left or right unless one first specifies the underlying subgroup. In other words: a right coset of one subgroup equals a left coset of a different (conjugate) subgroup. If the left cosets and right cosets are the same, then H is a normal subgroup and the cosets form a group called the quotient or factor group.

The map gH ↦ (gH)−1 = Hg−1 defines a bijection between the left cosets and the right cosets of H, so the number of left cosets is equal to the number of right cosets. The common value is called the index of H in G.

For abelian groups, left cosets and right cosets are always the same. 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.

Examples[edit]

C2[edit]

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[edit]

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[edit]

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

\{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.

Definition using equivalence classes[edit]

Some authors[2] define the left cosets of H in G to be the equivalence classes under the equivalence relation on G given by x ~ y if and only if x−1yH. The relation can also be defined by x ~ y if and only if xh=y for some h in H. It can be shown that the relation given is, in fact, an equivalence relation and that the two definitions are equivalent. It follows that any two left cosets of H in G are either identical or disjoint. In other words every element of G belongs to one and only one left coset and so the left cosets form a partition of G.[3] Corresponding statements are true for right cosets.

Double cosets[edit]

Main article: Double coset

Given two subgroups, H and K of a group G, the double coset of H and K in G are 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.[4]

Notation[edit]

Let G be a group with subgroups H and K.

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

General properties[edit]

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[edit]

Main article: Index of a subgroup

All left cosets and all right cosets have the same order (number of elements, or cardinality in the case of an infinite H), equal to the order of H (because H is itself a coset). 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, as per the formula:

|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[edit]

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".

Applications[edit]

See also[edit]

References[edit]

  1. ^ Joshi p. 323
  2. ^ e.g. Zassenhaus
  3. ^ Joshi Corollary 2.3
  4. ^ Scott p. 19
  • Scott, W.R. (1987). "§1.7 Cosets and index". Group Theory. Courier Dover Publications. pp. 19 ff. ISBN 0-486-65377-3. 
  • Joshi, K. D. (1989). "§5.2 Cosets of Subgroups". Foundations of Discrete Mathematics. New Age International. pp. 322 ff. ISBN 81-224-0120-1. 
  • Zassenhaus, Hans J. (1999). "§1.4 Subgroups". The Theory of Groups. Courier Dover Publications. pp. 10 ff. ISBN 0-486-40922-8. 

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