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A quotient group or factor group is a mathematical group obtained by aggregating similar elements of a larger group using an
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 ...
that preserves some of the group structure (the rest of the structure is "factored" out). For example, the cyclic group of addition modulo ''n'' can be obtained from the group of
integer An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign ( −1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the languag ...
s under addition by identifying elements that differ by a multiple of n and defining a group structure that operates on each such class (known as a congruence class) as a single entity. It is part of the mathematical field known as
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 ...
. For a congruence relation on a group, the
equivalence class 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 a ...
of the identity element is always a normal subgroup of the original group, and the other equivalence classes are precisely the cosets of that normal subgroup. The resulting quotient is written G\,/\,N, where G is the original group and N is the normal subgroup. (This is pronounced G\bmod N, where \mbox is short for
modulo In computing, the modulo operation returns the remainder or signed remainder of a division, after one number is divided by another (called the '' modulus'' of the operation). Given two positive numbers and , modulo (often abbreviated as ) is ...
.) Much of the importance of quotient groups is derived from their relation to
homomorphisms In algebra, a homomorphism is a structure-preserving map between two algebraic structures of the same type (such as two groups, two rings, or two vector spaces). The word ''homomorphism'' comes from the Ancient Greek language: () meaning "same ...
. The first isomorphism theorem states that the
image An image is a visual representation of something. It can be two-dimensional, three-dimensional, or somehow otherwise feed into the visual system to convey information. An image can be an artifact, such as a photograph or other two-dimensio ...
of any group ''G'' under a homomorphism is always
isomorphic In mathematics, an isomorphism is a structure-preserving mapping between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between them. The word i ...
to a quotient of G. Specifically, the image of G under a homomorphism \varphi: G \rightarrow H is isomorphic to G\,/\,\ker(\varphi) where \ker(\varphi) denotes the kernel of \varphi. The dual notion of a quotient group is a subgroup, these being the two primary ways of forming a smaller group from a larger one. Any normal subgroup has a corresponding quotient group, formed from the larger group by eliminating the distinction between elements of the subgroup. In
category theory Category theory is a general theory of mathematical structures and their relations that was introduced by Samuel Eilenberg and Saunders Mac Lane in the middle of the 20th century in their foundational work on algebraic topology. Nowadays, ca ...
, quotient groups are examples of quotient objects, which are dual to subobjects.


Definition and illustration

Given a group G and a subgroup H, and an element a \in G, one can consider the corresponding left coset: aH := \left\. Cosets are a natural class of subsets of a group; for example consider the
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 ...
''G'' of
integer An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign ( −1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the languag ...
s, with
operation Operation or Operations may refer to: Arts, entertainment and media * ''Operation'' (game), a battery-operated board game that challenges dexterity * Operation (music), a term used in musical set theory * ''Operations'' (magazine), Multi-Man ...
defined by the usual addition, and the subgroup H of even integers. Then there are exactly two cosets: 0+H, which are the even integers, and 1+H, which are the odd integers (here we are using additive notation for the binary operation instead of multiplicative notation). For a general subgroup ''H'', it is desirable to define a compatible group operation on the set of all possible cosets, \left\. This is possible exactly when ''H'' is a normal subgroup, see below. A subgroup N of a group ''G'' is normal
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 ...
the coset equality aN = Na holds for all a \in G. A normal subgroup of ''G'' is denoted N.


Definition

Let ''N'' be a normal subgroup of a group ''G'' . Define the set G\,/\,N to be the set of all left cosets of ''N'' in ''G'' . That is, G\,/\,N = \left\. Since the identity element e \in N, a \in aN. Define a binary operation on the set of cosets, G\,/\,N, as follows. For each aN and bN in G\,/\,N, the product of aN and bN, (aN)(bN), is (ab)N. This works only because (ab)N does not depend on the choice of the representatives, a and b, of each left coset, aN and bN. To prove this, suppose xN = aN and yN = bN for some x, y, a, b \in G. Then :(ab)N = a(bN) = a(yN) = a(Ny) = (aN)y = (xN)y = x(Ny) = x(yN) = (xy)N. This depends on the fact that ''N'' is a normal subgroup. It still remains to be shown that this condition is not only sufficient but necessary to define the operation on ''G''/''N''. To show that it is necessary, consider that for a subgroup ''N'' of ''G'', we have been given that the operation is well defined. That is, for all xN = aN and yN = bN'','' for x, y, a, b \in G, \; (ab)N = (xy)N. Let n \in N and g \in G. Since eN = nN'','' we have gN = (eg)N = (eN)(gN) = (nN)(gN) = (ng)N. Now, gN = (ng)N \Leftrightarrow N = (g^ng)N \Leftrightarrow g^ng \in N, \; \forall \, n \in N and g \in G. Hence ''N'' is a normal subgroup of ''G'' . It can also be checked that this operation on G\,/\,N is always associative, G\,/\,N has identity element ''N'', and the inverse of element aN can always be represented by a^N. Therefore, the set G\,/\,N together with the operation defined by (aN)(bN) = (ab)N forms a group, the quotient group of ''G'' by ''N''. Due to the normality of ''N'', the left cosets and right cosets of ''N'' in ''G'' are the same, and so, G\,/\,N could have been defined to be the set of right cosets of ''N'' in ''G'' .


Example: Addition modulo 6

For example, consider the group with addition modulo 6: G = \left\. Consider the subgroup ''N = \left\'', which is normal because ''G'' 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 ...
. Then the set of (left) cosets is of size three: : G\,/\,N = \left\ = \left\ = \left\. The binary operation defined above makes this set into a group, known as the quotient group, which in this case is isomorphic to the cyclic group of order 3.


Motivation for the name "quotient"

The reason G\,/\,N is called a quotient group comes from division of
integer An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign ( −1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the languag ...
s. When dividing 12 by 3 one obtains the answer 4 because one can regroup 12 objects into 4 subcollections of 3 objects. The quotient group is the same idea, although we end up with a group for a final answer instead of a number because groups have more structure than an arbitrary collection of objects. To elaborate, when looking at G\,/\,N with ''N'' a normal subgroup of ''G'', the group structure is used to form a natural "regrouping". These are the cosets of ''N'' in ''G''. Because we started with a group and normal subgroup, the final quotient contains more information than just the number of cosets (which is what regular division yields), but instead has a group structure itself.


Examples


Even and odd integers

Consider the group of
integer An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign ( −1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the languag ...
s \Z (under addition) and the subgroup 2\Z consisting of all even integers. This is a normal subgroup, because \Z 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 ...
. There are only two cosets: the set of even integers and the set of odd integers, and therefore the quotient group \Z\,/\,2\Z is the cyclic group with two elements. This quotient group is isomorphic with the set \left\ with addition modulo 2; informally, it is sometimes said that \Z\,/\,2\Z ''equals'' the set \left\ with addition modulo 2. Example further explained... : Let \gamma(m) be the remainders of m \in \Z when dividing by 2 . Then, \gamma(m)=0 when m is even and \gamma(m)=1 when m is odd. : By definition of \gamma , the kernel of \gamma , \ker(\gamma) = \ , is the set of all even integers. : Let H= \ker(\gamma). Then, H is a subgroup, because the identity in \Z , which is 0 , is in H , the sum of two even integers is even and hence if m and n are in H , m+n is in H (closure) and if m is even, -m is also even and so H contains its inverses. : Define \mu : \to \Z_2 as \mu(aH)=\gamma(a) for a\in\Z and is the quotient group of left cosets; =\ . : Note that we have defined \mu , \mu(aH) is 1 if a is odd and 0 if a is even. : Thus, \mu is an isomorphism from to \Z_2 .


Remainders of integer division

A slight generalization of the last example. Once again consider the group of integers \Z under addition. Let ''n'' be any positive integer. We will consider the subgroup n\Z of \Z consisting of all multiples of ''n''. Once again n\Z is normal in \Z because \Z is abelian. The cosets are the collection \left\. An integer ''k'' belongs to the coset r+n\Z, where ''r'' is the remainder when dividing ''k'' by ''n''. The quotient \Z\,/\,n\Z can be thought of as the group of "remainders" modulo n. This is a cyclic group of order ''n''.


Complex integer roots of 1

The twelfth roots of unity, which are points on the complex unit circle, form a multiplicative abelian group ''G'', shown on the picture on the right as colored balls with the number at each point giving its complex argument. Consider its subgroup ''N'' made of the fourth roots of unity, shown as red balls. This normal subgroup splits the group into three cosets, shown in red, green and blue. One can check that the cosets form a group of three elements (the product of a red element with a blue element is blue, the inverse of a blue element is green, etc.). Thus, the quotient group ''G\,/\,N'' is the group of three colors, which turns out to be the cyclic group with three elements.


The real numbers modulo the integers

Consider the group of
real number In mathematics, a real number is a number that can be used to measure a ''continuous'' one-dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every ...
s \R under addition, and the subgroup \Z of integers. Each coset of \Z in \R is a set of the form a+\Z, where ''a'' is a real number. Since a_1+\Z and a_2+\Z are identical sets when the non- integer parts of ''a_1'' and ''a_2'' are equal, one may impose the restriction 0 \leq a < 1 without change of meaning. Adding such cosets is done by adding the corresponding real numbers, and subtracting 1 if the result is greater than or equal to 1. The quotient group \R\,/\,\Z is isomorphic to the circle group, the group of
complex number In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the fo ...
s of
absolute value In mathematics, the absolute value or modulus of a real number x, is the non-negative value without regard to its sign. Namely, , x, =x if is a positive number, and , x, =-x if x is negative (in which case negating x makes -x positive), ...
1 under multiplication, or correspondingly, the group of
rotation Rotation, or spin, is the circular movement of an object around a '' central axis''. A two-dimensional rotating object has only one possible central axis and can rotate in either a clockwise or counterclockwise direction. A three-dimensional ...
s in 2D about the origin, that is, the special
orthogonal group In mathematics, the orthogonal group in dimension , denoted , is the group of distance-preserving transformations of a Euclidean space of dimension that preserve a fixed point, where the group operation is given by composing transformations. ...
\mbox(2). An isomorphism is given by f(a+\Z) = \exp(2\pi ia) (see
Euler's identity In mathematics, Euler's identity (also known as Euler's equation) is the equality e^ + 1 = 0 where : is Euler's number, the base of natural logarithms, : is the imaginary unit, which by definition satisfies , and : is pi, the ratio of the circ ...
).


Matrices of real numbers

If ''G'' is the group of invertible 3 \times 3 real matrices, and ''N'' is the subgroup of 3 \times 3 real matrices with
determinant In mathematics, the determinant is a scalar value that is a function of the entries of a square matrix. It characterizes some properties of the matrix and the linear map represented by the matrix. In particular, the determinant is nonzero if a ...
1, then ''N'' is normal in ''G'' (since it is the kernel of the determinant
homomorphism In algebra, a homomorphism is a morphism, structure-preserving map (mathematics), map between two algebraic structures of the same type (such as two group (mathematics), groups, two ring (mathematics), rings, or two vector spaces). The word ''homo ...
). The cosets of ''N'' are the sets of matrices with a given determinant, and hence ''G\,/\,N'' is isomorphic to the multiplicative group of non-zero real numbers. The group ''N'' is known as the special linear group \mbox(3).


Integer modular arithmetic

Consider the abelian group \Z_4 = \Z\,/\,4 \Z (that is, the set \left\ with addition
modulo In computing, the modulo operation returns the remainder or signed remainder of a division, after one number is divided by another (called the '' modulus'' of the operation). Given two positive numbers and , modulo (often abbreviated as ) is ...
4), and its subgroup \left\. The quotient group \Z_4\,/\,\left\ is \left\. This is a group with identity element \left\, and group operations such as \left\ + \left\ = \left\. Both the subgroup \left\ and the quotient group \left\ are isomorphic with \Z_2.


Integer multiplication

Consider the multiplicative group G=(\Z_)^. The set ''N'' of nth residues is a multiplicative subgroup isomorphic to (\Z_)^. Then ''N'' is normal in ''G'' and the factor group ''G\,/\,N'' has the cosets N, (1+n)N, (1+n)2N, \;\ldots, (1+n)n-1N. The Paillier cryptosystem is based on the
conjecture In mathematics, a conjecture is a conclusion or a proposition that is proffered on a tentative basis without proof. Some conjectures, such as the Riemann hypothesis (still a conjecture) or Fermat's Last Theorem (a conjecture until proven in ...
that it is difficult to determine the coset of a random element of ''G'' without knowing the factorization of ''n''.


Properties

The quotient group G\,/\,G is
isomorphic In mathematics, an isomorphism is a structure-preserving mapping between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between them. The word i ...
to the trivial group (the group with one element), and G\,/\,\left\ is isomorphic to ''G''. The
order Order, ORDER or Orders may refer to: * Categorization, the process in which ideas and objects are recognized, differentiated, and understood * Heterarchy, a system of organization wherein the elements have the potential to be ranked a number of ...
of ''G\,/\,N'', by definition the number of elements, is equal to \vert G : N \vert, the index of ''N'' in ''G''. If ''G'' is finite, the index is also equal to the order of ''G'' divided by the order of ''N''. The set ''G\,/\,N'' may be finite, although both ''G'' and ''N'' are infinite (for example, \Z\,/\,2\Z). There is a "natural"
surjective In mathematics, a surjective function (also known as surjection, or onto function) is a function that every element can be mapped from element so that . In other words, every element of the function's codomain is the image of one element o ...
group homomorphism \pi: G \rightarrow G\,/\,N, sending each element g of ''G'' to the coset of ''N'' to which ''g'' belongs, that is: \pi(g) = gN. The mapping \pi is sometimes called the ''canonical projection of G onto G\,/\,N''. Its kernel is ''N''. There is a bijective correspondence between the subgroups of ''G'' that contain ''N'' and the subgroups of ''G\,/\,N''; if H is a subgroup of ''G'' containing ''N'', then the corresponding subgroup of ''G\,/\,N'' is \pi(H). This correspondence holds for normal subgroups of ''G'' and ''G\,/\,N'' as well, and is formalized in the lattice theorem. Several important properties of quotient groups are recorded in the fundamental theorem on homomorphisms and the
isomorphism theorem In mathematics, specifically abstract algebra, the isomorphism theorems (also known as Noether's isomorphism theorems) are theorems that describe the relationship between quotients, homomorphisms, and subobjects. Versions of the theorems exis ...
s. If ''G'' 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 ...
,
nilpotent In mathematics, an element x of a ring R is called nilpotent if there exists some positive integer n, called the index (or sometimes the degree), such that x^n=0. The term was introduced by Benjamin Peirce in the context of his work on the cl ...
, solvable,
cyclic Cycle, cycles, or cyclic may refer to: Anthropology and social sciences * Cyclic history, a theory of history * Cyclical theory, a theory of American political history associated with Arthur Schlesinger, Sr. * Social cycle, various cycles in so ...
or finitely generated, then so is ''G\,/\,N''. If ''H'' is a subgroup in a finite group ''G'', and the order of ''H'' is one half of the order of ''G'', then ''H'' is guaranteed to be a normal subgroup, so ''G\,/\,H'' exists and is isomorphic to C_2. This result can also be stated as "any subgroup of index 2 is normal", and in this form it applies also to infinite groups. Furthermore, if p is the smallest prime number dividing the order of a finite group, ''G'', then if ''G\,/\,H'' has order ''p'', ''H'' must be a normal subgroup of ''G''. Given ''G'' and a normal subgroup ''N'', then ''G'' is a group extension of ''G\,/\,N'' by ''N''. One could ask whether this extension is trivial or split; in other words, one could ask whether ''G'' is a
direct product In mathematics, one can often define a direct product of objects already known, giving a new one. This generalizes the Cartesian product of the underlying sets, together with a suitably defined structure on the product set. More abstractly, one t ...
or
semidirect product In mathematics, specifically in group theory, the concept of a semidirect product is a generalization of a direct product. There are two closely related concepts of semidirect product: * an ''inner'' semidirect product is a particular way in wh ...
of ''N'' and ''G\,/\,N''. This is a special case of the extension problem. An example where the extension is not split is as follows: Let G = \Z_4 = \left\, and N = \left\, which is isomorphic to \Z_2. Then ''G\,/\,N'' is also isomorphic to \Z_2. But \Z_2 has only the trivial
automorphism In mathematics, an automorphism is an isomorphism from a mathematical object to itself. It is, in some sense, a symmetry of the object, and a way of mapping the object to itself while preserving all of its structure. The set of all automorphis ...
, so the only semi-direct product of ''N'' and ''G\,/\,N'' is the direct product. Since \Z_4 is different from \Z_2 \times \Z_2, we conclude that ''G'' is not a semi-direct product of ''N'' and ''G\,/\,N''.


Quotients of Lie groups

If ''G'' is a
Lie group In mathematics, a Lie group (pronounced ) is a group that is also a differentiable manifold. A manifold is a space that locally resembles Euclidean space, whereas groups define the abstract concept of a binary operation along with the addi ...
and ''N'' is a normal and closed (in the topological rather than the algebraic sense of the word)
Lie subgroup In mathematics, a Lie group (pronounced ) is a group that is also a differentiable manifold. A manifold is a space that locally resembles Euclidean space, whereas groups define the abstract concept of a binary operation along with the additi ...
of ''G'', the quotient is also a Lie group. In this case, the original group ''G'' has the structure of a
fiber bundle In mathematics, and particularly topology, a fiber bundle (or, in Commonwealth English: fibre bundle) is a space that is a product space, but may have a different topological structure. Specifically, the similarity between a space E and a ...
(specifically, a principal ''N''-bundle), with base space and fiber ''N''. The dimension of equals \dim G - \dim N.John M. Lee, Introduction to Smooth Manifolds, Second Edition, theorem 21.17 Note that the condition that ''N'' is closed is necessary. Indeed, if ''N'' is not closed then the quotient space is not a T1-space (since there is a coset in the quotient which cannot be separated from the identity by an open set), and thus not a
Hausdorff space In topology and related branches of mathematics, a Hausdorff space ( , ), separated space or T2 space is a topological space where, for any two distinct points, there exist neighbourhoods of each which are disjoint from each other. Of the ma ...
. For a non-normal Lie subgroup ''N'', the space G\,/\,N of left cosets is not a group, but simply a
differentiable manifold In mathematics, a differentiable manifold (also differential manifold) is a type of manifold that is locally similar enough to a vector space to allow one to apply calculus. Any manifold can be described by a collection of charts (atlas). One ma ...
on which ''G'' acts. The result is known as a
homogeneous space In mathematics, particularly in the theories of Lie groups, algebraic groups and topological groups, a homogeneous space for a group ''G'' is a non-empty manifold or topological space ''X'' on which ''G'' acts transitively. The elements ...
.


See also

* Group extension *
Quotient category In mathematics, a quotient category is a category obtained from another one by identifying sets of morphisms. Formally, it is a quotient object in the category of (locally small) categories, analogous to a quotient group or quotient space, but ...
* Short exact sequence


Notes


References

* * {{citation , last1=Herstein , first1=I. N. , year=1975 , title=Topics in Algebra , edition=2nd , publisher= Wiley , location=New York , isbn=0-471-02371-X Group theory Group