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In
mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and their changes (cal ...
, specifically
group theory In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no ...
, a
subgroup In group theory, a branch of mathematics, given a group (mathematics), 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 ...
of a
group A group is a number A number is a mathematical object used to counting, count, measurement, measure, and nominal number, label. The original examples are the natural numbers 1, 2, 3, 4, and so forth. Numbers can be represented in language with ...
may be used to decompose the underlying set of into
disjoint
disjoint
equal-size subsets called cosets. There are ''left cosets'' and ''right cosets''. Cosets (both left and right) have the same number of elements (
cardinality In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
) 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 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 megastructure in the ''Halo'' series ...
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 In abstract algebra In algebra, which is a broad division of mathematics, abstract algebra (occasionally called modern algebra) is the study of algebraic structures. Algebraic structures include group (mathematics), groups, ring (mathematics), ...
, 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 In algebra, which is a broad division of mathematics, abstract algebra (occasionally called modern algebra) is the study of algebraic structures. Algebraic structures include group (mathematics), groups, ring (mathematics), ...
) 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 Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...
s and
error-correcting code In computing Computing is any goal-oriented activity requiring, benefiting from, or creating computing machinery. It includes the study and experimentation of algorithmic processes and development of both computer hardware , hardware and softwa ...
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, 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 British may refer to: Peoples, culture, and language * British people, nationals or natives of the United Kingdom, British Overseas Territories, and Crown Dependencies. ** Britishness, the British identity a ...
: 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 Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...
. Define two elements of , and , to be equivalent with respect to the subgroup if (or equivalently if belongs to ). The
equivalence classes In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). I ...
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 Representative may refer to: Politics *Representative democracy, type of democracy in which elected officials represent a group of people *House of Representatives, legislative body in various countries or sub-national entities *Legislator, someone ...
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 Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities an ...
, 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 with respect to , that is, .


Normal subgroups

A subgroup of a group is a
normal subgroup In abstract algebra In algebra, which is a broad division of mathematics, abstract algebra (occasionally called modern algebra) is the study of algebraic structures. Algebraic structures include group (mathematics), groups, ring (mathematics), ...
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 Center or centre may refer to: Mathematics *Center (geometry) In geometry, a centre (or center) (from Ancient Greek language, Greek ''κέντρον'') of an object is a point in some sense in the middle of the object. According to the speci ...
of , then .) On the other hand, if the subgroup is normal the set of all cosets form 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 Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
in the case of an
infinite Infinite may refer to: Mathematics *Infinite set, a set that is not a finite set *Infinity, an abstract concept describing something without any limit Music *Infinite (band), a South Korean boy band *''Infinite'' (EP), debut EP of American musi ...

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 An additive group is a group of which the group operation is to be thought of as ''addition'' in some sense. It is usually abelian, and typically written using the symbol + for its binary operation. This terminology is widely used with structure ...
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 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 In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
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 Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...
s. The elements (vectors) of a vector space form an
abelian group In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities an ...
under
vector addition In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). I ...

vector addition
. The subspaces of the vector space are
subgroups In group theory In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ...
of this group. For a vector space , a subspace , and a fixed vector in , the sets \ are called
affine subspace In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...

affine subspace
s, and are cosets (both left and right, since the group is abelian). In terms of 3-dimensional
geometric Geometry (from the grc, γεωμετρία; '' geo-'' "earth", '' -metron'' "measurement") is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space that are related with distance, shape, size, ...
vectors, these affine subspaces are all the "lines" or "planes"
parallel Parallel may refer to: Computing * Parallel algorithm In computer science Computer science deals with the theoretical foundations of information, algorithms and the architectures of its computation as well as practical techniques for their a ...
to the subspace, which is a line or plane going through the origin. For example, consider the
plane Plane or planes may refer to: * Airplane An airplane or aeroplane (informally plane) is a fixed-wing aircraft A fixed-wing aircraft is a heavier-than-air flying machine Early flying machines include all forms of aircraft studied ...
. 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 ACTION is a bus operator in , Australia owned by the . History On 19 July 1926, the commenced operating public bus services between Eastlake (now ) in the south and in the north. The service was first known as Canberra City Omnibus Se ...
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 physical body, object such as the trajectory of a planet around a star, or of a natural satellite around a planet, or of an satellite, artificial satellite around an object or po ...
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 '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 language, German origin, derived from the noun meaning "weaving, weaver". In some cases, following migration to English-speaking countries, it has been anglicised to the English surname 'Webber' or ev ...
) and ''conjugate group'' ( Burnside). Galois was concerned with deciding when a given
polynomial equation In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It h ...
was solvable by radicals. A tool that he developed was in noting that a subgroup of a group of
permutation In , a permutation of a is, loosely speaking, an arrangement of its members into a or , or if the set is already ordered, a rearrangement of its elements. The word "permutation" also refers to the act or process of changing the linear order o ...

permutation
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 Marie Ennemond Camille Jordan (; 5 January 1838 – 22 January 1922) was a French mathematician, known both for his foundational work in group theory and for his influential ''Cours d'analyse''. Biography Jordan was born in Lyon and educated a ...
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 arrayIn coding theory, a standard array (or Slepian array) is a q^ by q^ array that lists all elements of a particular \mathbb_q^n vector space. Standard arrays are used to Decoding methods, decode linear codes; i.e. to find the corresponding codeword for ...
. 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 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 In coding theory, decoding is the process of translating received messages into codewords of a given code. There have been many common methods of mapping messages to codewords. These are often used to recover messages sent over a noisy channel, such ...
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 Logic is an interdisciplinary field which studies truth and reasoning. Informal logic seeks to characterize Validity (logic), valid arguments informally, for instance by listing varieties of fallacies. Formal logic represents st ...
is in . The vector is called the ''syndrome'' of , and by
linearity Linearity is the property of a mathematical relationship (''function (mathematics), function'') that can be graph of a function, graphically represented as a straight Line (geometry), line. Linearity is closely related to ''Proportionality (math ...

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 set In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and th ...
s, a type of
non-measurable set In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...
. *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 3-D combination puzzle invented in 1974 by Hungarian sculptor and professor of architecture Ernő Rubik. Originally called the Magic Cube, the puzzle was licensed by Rubik to be sold by Ideal Toy Company, Ideal Toy Corp. ...

Rubik's Cube
relies heavily on cosets. *In geometry, a Clifford–Klein form 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 (mathematics), space is a group homomorphism of a given group (mathematics), group into the group of transformation (geometry), transformations of the space. Similarly, a group action on a mathematical ...
on the
homogeneous space In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...
.


See also

*
Heap Heap or HEAP may refer to: Computing and mathematics * Heap (data structure), a data structure commonly used to implement a priority queue * Heap (mathematics), a generalization of a group * Heap (programming) (or free store), an area of memory for ...
* Coset enumeration


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