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mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, specifically
group theory In abstract algebra, group theory studies the algebraic structures known as groups. The concept of a group is central to abstract algebra: other well-known algebraic structures, such as rings, fields, and vector spaces, can all be seen ...
, a subgroup of a group may be used to decompose the underlying
set Set, The Set, SET or SETS may refer to: Science, technology, and mathematics Mathematics *Set (mathematics), a collection of elements *Category of sets, the category whose objects and morphisms are sets and total functions, respectively Electro ...
of into disjoint, equal-size
subset In mathematics, set ''A'' is a subset of a set ''B'' if all elements of ''A'' are also elements of ''B''; ''B'' is then a superset of ''A''. It is possible for ''A'' and ''B'' to be equal; if they are unequal, then ''A'' is a proper subset of ...
s called cosets. There are ''left cosets'' and ''right cosets''. Cosets (both left and right) have the same number of elements (
cardinality In mathematics, the cardinality of a set is a measure of the number of elements of the set. For example, the set A = \ contains 3 elements, and therefore A has a cardinality of 3. Beginning in the late 19th century, this concept was generalized ...
) as does . Furthermore, itself is both a left coset and a right coset. The number of left cosets of in is equal to the number of right cosets of in . This common value is called the
index Index (or its plural form indices) may refer to: Arts, entertainment, and media Fictional entities * Index (''A Certain Magical Index''), a character in the light novel series ''A Certain Magical Index'' * The Index, an item on a Halo megastru ...
of in and is usually denoted by . 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 Finite is the opposite of infinite. It may refer to: * Finite number (disambiguation) * Finite set, a set whose cardinality (number of elements) is some natural number * Finite verb, a verb form that has a subject, usually being inflected or ma ...
, the number of elements of every subgroup of divides the number of elements of . Cosets of a particular type of subgroup (a
normal subgroup In abstract algebra, a normal subgroup (also known as an invariant subgroup or self-conjugate subgroup) is a subgroup that is invariant under conjugation by members of the group of which it is a part. In other words, a subgroup N of the group G ...
) can be used as the elements of another group called a quotient group or factor group. Cosets also appear in other areas of mathematics such as
vector space In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called '' vectors'', may be added together and multiplied ("scaled") by numbers called ''scalars''. Scalars are often real numbers, but can ...
s and
error-correcting code In computing, telecommunication, information theory, and coding theory, an error correction code, sometimes error correcting code, (ECC) is used for controlling errors in data over unreliable or noisy communication channels. The central idea i ...
s.


Definition

Let be a subgroup of the group whose operation is written multiplicatively (juxtaposition denotes the group operation). Given an element of , the left cosets of in are the sets obtained by multiplying each element of by a fixed element of (where is the left factor). In symbols these are, The right cosets are defined similarly, except that the element is now a right factor, that is, As varies through the group, it would appear that many cosets (right or left) would be generated. Nevertheless, it turns out that any two left cosets (respectively right cosets) are either disjoint or are identical as sets. If the group operation is written additively, as is often the case when the group is
abelian Abelian may refer to: Mathematics Group theory * Abelian group, a group in which the binary operation is commutative ** Category of abelian groups (Ab), has abelian groups as objects and group homomorphisms as morphisms * Metabelian group, a grou ...
, the notation used changes to or , respectively.


First example

Let be the dihedral group of order six. Its elements may be represented by . In this group, and . This is enough information to fill in the entire
Cayley table Named after the 19th century British mathematician Arthur Cayley, a Cayley table describes the structure of a finite group by arranging all the possible products of all the group's elements in a square table reminiscent of an addition or multiplic ...
: Let be the subgroup . The (distinct) left cosets of are: *, *, and *. Since all the elements of have now appeared in one of these cosets, generating any more can not give new cosets, since a new coset would have to have an element in common with one of these and therefore be identical to one of these cosets. For instance, . The right cosets of are: *, * , and *. In this example, except for , no left coset is also a right coset. Let be the subgroup . The left cosets of are and . The right cosets of are and . In this case, every left coset of is also a right coset of . Let be a subgroup of a group and suppose that , . The following statements are equivalent: * * * * *


Properties

The disjointness of non-identical cosets is a result of the fact that if belongs to then . For if then there must exist an such that . Thus . Moreover, since is a group, left multiplication by is a bijection, and . Thus every element of belongs to exactly one left coset of the subgroup , and is itself a left coset (and the one that contains the identity). Two elements being in the same left coset also provide a natural
equivalence relation In mathematics, an equivalence relation is a binary relation that is reflexive, symmetric and transitive. The equipollence relation between line segments in geometry is a common example of an equivalence relation. Each equivalence relatio ...
. Define two elements of , and , to be equivalent with respect to the subgroup if (or equivalently if belongs to ). The
equivalence classes In mathematics, when the elements of some set S have a notion of equivalence (formalized as an equivalence relation), then one may naturally split the set S into equivalence classes. These equivalence classes are constructed so that elements ...
of this relation are the left cosets of . As with any set of equivalence classes, they form a partition of the underlying set. 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. Similar statements apply to right cosets. If is an
abelian group In mathematics, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on the order in which they are written. That is, the group operation is comm ...
, then for every subgroup of and every element of . For general groups, given an element and a subgroup of a group , the right coset of with respect to is also the left coset of the
conjugate subgroup In mathematics, especially group theory, two elements a and b of a group are conjugate if there is an element g in the group such that b = gag^. This is an equivalence relation whose equivalence classes are called conjugacy classes. In other ...
with respect to , that is, .


Normal subgroups

A subgroup of a group is a
normal subgroup In abstract algebra, a normal subgroup (also known as an invariant subgroup or self-conjugate subgroup) is a subgroup that is invariant under conjugation by members of the group of which it is a part. In other words, a subgroup N of the group G ...
of if and only if for all elements of the corresponding left and right cosets are equal, that is, . This is the case for the subgroup in the first example above. Furthermore, the cosets of in form a group called the quotient group or factor group . If is not normal in , then its left cosets are different from its right cosets. That is, there is an in such that no element satisfies . This means that the partition of into the left cosets of is a different partition than the partition of into right cosets of . This is illustrated by the subgroup in the first example above. (''Some'' cosets may coincide. For example, if is in the center of , then .) On the other hand, if the subgroup is normal the set of all cosets forms a group called the quotient group with the operation defined by . Since every right coset is a left coset, there is no need to distinguish "left cosets" from "right cosets".


Index of a subgroup

Every left or right coset of has the same number of elements (or
cardinality In mathematics, the cardinality of a set is a measure of the number of elements of the set. For example, the set A = \ contains 3 elements, and therefore A has a cardinality of 3. Beginning in the late 19th century, this concept was generalized ...
in the case of an infinite ) as itself. Furthermore, the number of left cosets is equal to the number of right cosets and is known as the index of in ''G'', written as . Lagrange's theorem allows us to compute the index in the case where and are finite: , G, = : HH, . This equation also holds in the case where the groups are infinite, although the meaning may be less clear.


More examples


Integers

Let be the additive group of the integers, and the subgroup . Then the cosets of in are the three sets , , and , where . These three sets partition the set , so there are no other right cosets of . Due to the
commutivity In mathematics, a binary operation is commutative if changing the order of the operands does not change the result. It is a fundamental property of many binary operations, and many mathematical proofs depend on it. Most familiar as the name of ...
of addition and . That is, every left coset of is also a right coset, so is a normal subgroup. (The same argument shows that every subgroup of an Abelian group is normal.) This example may be generalized. Again let be the additive group of the integers, , and now let the subgroup , where is a positive integer. Then the cosets of in are the sets , , ..., , where . There are no more than cosets, because . The coset is the congruence class of modulo . The subgroup is normal in , and so, can be used to form the quotient group the group of integers mod ''m''.


Vectors

Another example of a coset comes from the theory of
vector space In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called '' vectors'', may be added together and multiplied ("scaled") by numbers called ''scalars''. Scalars are often real numbers, but can ...
s. The elements (vectors) of a vector space form an
abelian group In mathematics, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on the order in which they are written. That is, the group operation is comm ...
under vector addition. The subspaces of the vector space are
subgroups In group theory, a branch of mathematics, given a group ''G'' under a binary operation ∗, a subset ''H'' of ''G'' is called a subgroup of ''G'' if ''H'' also forms a group under the operation ∗. More precisely, ''H'' is a subgrou ...
of this group. For a vector space , a subspace , and a fixed vector in , the sets \ are called affine subspaces, and are cosets (both left and right, since the group is abelian). In terms of 3-dimensional
geometric Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is ca ...
vectors, these affine subspaces are all the "lines" or "planes"
parallel Parallel is a geometric term of location which may refer to: Computing * Parallel algorithm * Parallel computing * Parallel metaheuristic * Parallel (software), a UNIX utility for running programs in parallel * Parallel Sysplex, a cluster o ...
to the subspace, which is a line or plane going through the origin. For example, consider the plane . If is a line through the origin , then is a subgroup of the abelian group . If is in , then the coset is a line parallel to and passing through .


Matrices

Let be the multiplicative group of matrices, G = \left \, and the subgroup of , H= \left \. For a fixed element of consider the left coset \begin \begin a & 0 \\ b & 1 \end H = &~ \left \ \\ =&~ \left \ \\ =&~ \left \. \end That is, the left cosets consist of all the matrices in having the same upper-left entry. This subgroup is normal in , but the subgroup T= \left \ is not normal in .


As orbits of a group action

A subgroup of a group can be used to define an action of on in two natural ways. A ''right action'', given by or a ''left action'', given by . The
orbit In celestial mechanics, an orbit is the curved trajectory of an object such as the trajectory of a planet around a star, or of a natural satellite around a planet, or of an artificial satellite around an object or position in space such as ...
of under the right action is the left coset , while the orbit under the left action is the right coset .


History

The concept of a coset dates back to Galois's work of 1830–31. He introduced a notation but did not provide a name for the concept. The term "co-set" appears for the first time in 1910 in a paper by G. A. Miller in the ''Quarterly Journal of Mathematics'' (vol. 41, p. 382). Various other terms have been used including the German ''Nebengruppen'' (
Weber Weber (, or ; German: ) is a surname of German origin, derived from the noun meaning " weaver". In some cases, following migration to English-speaking countries, it has been anglicised to the English surname 'Webber' or even 'Weaver'. Notable pe ...
) and ''conjugate group'' ( Burnside). Galois was concerned with deciding when a given
polynomial equation In mathematics, an algebraic equation or polynomial equation is an equation of the form :P = 0 where ''P'' is a polynomial with coefficients in some field (mathematics), field, often the field of the rational numbers. For many authors, the term '' ...
was solvable by radicals. A tool that he developed was in noting that a subgroup of a group of
permutation In mathematics, a permutation of a set is, loosely speaking, an arrangement of its members into a sequence or linear order, or if the set is already ordered, a rearrangement of its elements. The word "permutation" also refers to the act or pro ...
s induced two decompositions of (what we now call left and right cosets). If these decompositions coincided, that is, if the left cosets are the same as the right cosets, then there was a way to reduce the problem to one of working over instead of . Camille Jordan in his commentaries on Galois's work in 1865 and 1869 elaborated on these ideas and defined normal subgroups as we have above, although he did not use this term. Calling the coset the ''left coset'' of with respect to , while most common today, has not been universally true in the past. For instance, would call a ''right coset'', emphasizing the subgroup being on the right.


An application from coding theory

A binary linear code is an -dimensional subspace of an -dimensional vector space over the binary field . As is an additive abelian group, is a subgroup of this group. Codes can be used to correct errors that can occur in transmission. When a ''codeword'' (element of ) is transmitted some of its bits may be altered in the process and the task of the receiver is to determine the most likely codeword that the corrupted ''received word'' could have started out as. This procedure is called ''decoding'' and if only a few errors are made in transmission it can be done effectively with only a very few mistakes. One method used for decoding uses an arrangement of the elements of (a received word could be any element of ) into a standard array. A standard array is a coset decomposition of put into tabular form in a certain way. Namely, the top row of the array consists of the elements of , written in any order, except that the zero vector should be written first. Then, an element of with a minimal number of ones that does not already appear in the top row is selected and the coset of containing this element is written as the second row (namely, the row is formed by taking the sum of this element with each element of directly above it). This element is called a
coset leader In coding theory, a coset leader is a word A word is a basic element of language that carries an objective or practical meaning, can be used on its own, and is uninterruptible. Despite the fact that language speakers often have an intuitive ...
and there may be some choice in selecting it. Now the process is repeated, a new vector with a minimal number of ones that does not already appear is selected as a new coset leader and the coset of containing it is the next row. The process ends when all the vectors of have been sorted into the cosets. An example of a standard array for the 2-dimensional code in the 5-dimensional space (with 32 vectors) is as follows: The decoding procedure is to find the received word in the table and then add to it the coset leader of the row it is in. Since in binary arithmetic adding is the same operation as subtracting, this always results in an element of . In the event that the transmission errors occurred in precisely the non-zero positions of the coset leader the result will be the right codeword. In this example, if a single error occurs, the method will always correct it, since all possible coset leaders with a single one appear in the array. Syndrome decoding can be used to improve the efficiency of this method. It is a method of computing the correct coset (row) that a received word will be in. For an -dimensional code in an -dimensional binary vector space, a parity check matrix is an matrix having the property that
if and only if In logic and related fields such as mathematics and philosophy, "if and only if" (shortened as "iff") is a biconditional logical connective between statements, where either both statements are true or both are false. The connective is bic ...
is in . The vector is called the ''syndrome'' of , and by linearity, every vector in the same coset will have the same syndrome. To decode, the search is now reduced to finding the coset leader that has the same syndrome as the received word.


Double cosets

Given two subgroups, and (which need not be distinct) of a group , the double cosets of and in are the sets of the form . These are the left cosets of and right cosets of when and respectively. Two double cosets and are either disjoint or identical. The set of all double cosets for fixed and form a partition of . A double coset contains the complete right cosets of (in ) of the form , with an element of and the complete left cosets of (in ) of the form , with in .


Notation

Let be a group with subgroups and . Several authors working with these sets have developed a specialized notation for their work, where * denotes the set of left cosets of in . * denotes the set of right cosets of in . * denotes the set of double cosets of and in , sometimes referred to as ''double coset space''. * denotes the double coset space of the subgroup in .


More applications

*Cosets of in are used in the construction of Vitali sets, a type of non-measurable set. *Cosets are central in the definition of the transfer. *Cosets are important in computational group theory. For example, Thistlethwaite's algorithm for solving
Rubik's Cube The Rubik's Cube is a Three-dimensional space, 3-D combination puzzle originally invented in 1974 by Hungarians, Hungarian sculptor and professor of architecture Ernő Rubik. Originally called the Magic Cube, the puzzle was licensed by Rubik t ...
relies heavily on cosets. *In geometry, a
Clifford–Klein form In mathematics, a Clifford–Klein form is a double coset space :Γ\''G''/''H'', where ''G'' is a reductive Lie group, ''H'' a closed subgroup of ''G'', and Γ a discrete subgroup of G that acts properly discontinuously on the homogeneous spac ...
is a double coset space , where is a reductive Lie group, is a closed subgroup, and is a discrete subgroup (of ) that acts
properly discontinuously In mathematics, a group action on a space is a group homomorphism of a given group into the group of transformations of the space. Similarly, a group action on a mathematical structure is a group homomorphism of a group into the automorphism ...
on the homogeneous space .


See also

* Heap *
Coset enumeration In mathematics, coset enumeration is the problem of counting the cosets of a subgroup ''H'' of a group ''G'' given in terms of a presentation. As a by-product, one obtains a permutation representation for ''G'' on the cosets of ''H''. If ''H'' has ...


Notes


References

* * * * * * * * *


Further reading

*


External links

* * * * *
Illustrated examples
*{{cite web, publisher=The Group Properties Wiki, work=groupprops, url=http://groupprops.subwiki.org/wiki/Coset, title=Coset Group theory de:Gruppentheorie#Nebenklassen