TheInfoList

OR:

A quotient group or factor group is a
math 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 ...
ematical 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 as ...
. For a congruence relation on a group, the equivalence class of the
identity element In mathematics, an identity element, or neutral element, of a binary operation operating on a set is an element of the set that leaves unchanged every element of the set when the operation is applied. This concept is used in algebraic structures s ...
is always a normal subgroup of the original group, and the other equivalence classes are precisely the
coset In mathematics, specifically group theory, a subgroup of a group may be used to decompose the underlying set of into disjoint, equal-size subsets called cosets. There are ''left cosets'' and ''right cosets''. Cosets (both left and right) ...
s 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.) Much of the importance of quotient groups is derived from their relation to homomorphisms. The
first 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 exist ...
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-dimension ...
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 $G\,/\,\ker\left(\varphi\right)$ where $\ker\left(\varphi\right)$ denotes the
kernel Kernel may refer to: Computing * Kernel (operating system), the central component of most operating systems * Kernel (image processing), a matrix used for image convolution * Compute kernel, in GPGPU programming * Kernel method, in machine learni ...
of $\varphi$. The dual notion of a quotient group is a
subgroup 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 ...
, 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, cat ...
, quotient groups are examples of
quotient object In category theory, a branch of mathematics, a subobject is, roughly speaking, an object that sits inside another object in the same category. The notion is a generalization of concepts such as subsets from set theory, subgroups from group theor ...
s, which are dual to
subobject In category theory, a branch of mathematics, a subobject is, roughly speaking, an object that sits inside another object in the same category. The notion is a generalization of concepts such as subsets from set theory, subgroups from group theor ...
s.

# Definition and illustration

Given a group $G$ and a subgroup $H$, and an element $a \in G$, one can consider the corresponding left
coset In mathematics, specifically group theory, a subgroup of a group may be used to decompose the underlying set of into disjoint, equal-size subsets called cosets. There are ''left cosets'' and ''right cosets''. Cosets (both left and right) ...
: $aH := \left\$. Cosets are a natural class of subsets of a group; for example consider the abelian group ''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-Ma ...
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 $bN$, $\left(aN\right)\left(bN\right)$, is $\left(ab\right)N$. This works only because $\left(ab\right)N$ 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 $x, y, a, b \in G, \; \left(ab\right)N = \left(xy\right)N$. Let $n \in N$ and $g \in G$. Since $eN = nN$'','' we have $gN = \left(eg\right)N = \left(eN\right)\left(gN\right) = \left(nN\right)\left(gN\right) = \left(ng\right)N$. Now, $gN = \left(ng\right)N \Leftrightarrow N = \left(g^ng\right)N \Leftrightarrow g^ng \in N, \; \forall \, n \in N$ 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 $\left(aN\right)\left(bN\right) = \left(ab\right)N$ 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. 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. 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 $\gamma\left(m\right)$ be the remainders of $m \in \Z$ when dividing by $2$. Then, $\gamma\left(m\right)=0$ when $m$ is even and $\gamma\left(m\right)=1$ when $m$ is odd. : By definition of $\gamma$, the kernel of $\gamma$, $\ker\left(\gamma\right)$ $= \$, is the set of all even integers. : Let $H=$ $\ker\left(\gamma\right)$. 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 $\mu\left(aH\right)=\gamma\left(a\right)$ for $a\in\Z$ and is the quotient group of left cosets; $=\$. : Note that we have defined $\mu$, $\mu\left(aH\right)$ 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 Complex commonly refers to: * Complexity, the behaviour of a system whose components interact in multiple ways so possible interactions are difficult to describe ** Complex system, a system composed of many components which may interact with each ...
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. Ever ...
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 In mathematics, the circle group, denoted by \mathbb T or \mathbb S^1, is the multiplicative group of all complex numbers with absolute value 1, that is, the unit circle in the complex plane or simply the unit complex numbers. \mathbb T = \. ...
, 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 form ...
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), an ...
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 $\mbox\left(2\right)$. An isomorphism is given by $f\left(a+\Z\right) = \exp\left(2\pi ia\right)$ (see Euler's identity).

## 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 an ...
1, then ''$N$'' is normal in ''$G$'' (since it is the
kernel Kernel may refer to: Computing * Kernel (operating system), the central component of most operating systems * Kernel (image processing), a matrix used for image convolution * Compute kernel, in GPGPU programming * Kernel method, in machine learni ...
of the determinant homomorphism). 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 In mathematics, the special linear group of degree ''n'' over a field ''F'' is the set of matrices with determinant 1, with the group operations of ordinary matrix multiplication and matrix inversion. This is the normal subgroup of the gener ...
$\mbox\left(3\right)$.

## Integer modular arithmetic

Consider the abelian group $\Z_4 = \Z\,/\,4 \Z$ (that is, the set $\left\$ with addition modulo 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 $G=\left(\Z_\right)^$. The set ''$N$'' of $n$th residues is a multiplicative subgroup isomorphic to $\left(\Z_\right)^$. Then ''$N$'' is normal in ''$G$'' and the factor group ''$G\,/\,N$'' has the cosets $N, \left(1+n\right)N, \left(1+n\right)2N, \;\ldots, \left(1+n\right)n-1N$. The
Paillier cryptosystem The Paillier cryptosystem, invented by and named after Pascal Paillier in 1999, is a probabilistic asymmetric algorithm for public key cryptography. The problem of computing ''n''-th residue classes is believed to be computationally difficult. The ...
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 1 ...
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 In mathematics, a trivial group or zero group is a group consisting of a single element. All such groups are isomorphic, so one often speaks of the trivial group. The single element of the trivial group is the identity element and so it is usually ...
(the group with one element), and $G\,/\,\left\$ is isomorphic to ''$G$''. The order of ''$G\,/\,N$'', by definition the number of elements, is equal to $\vert G : N \vert$, 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 ''$N$'' 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 of ...
group homomorphism $\pi: G \rightarrow G\,/\,N$, sending each element $g$ of ''$G$'' to the coset of ''$N$'' to which ''$g$'' belongs, that is: $\pi\left(g\right) = gN$. The mapping $\pi$ is sometimes called the ''canonical projection of $G$ onto $G\,/\,N$''. Its
kernel Kernel may refer to: Computing * Kernel (operating system), the central component of most operating systems * Kernel (image processing), a matrix used for image convolution * Compute kernel, in GPGPU programming * Kernel method, in machine learni ...
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 $\pi\left(H\right)$. 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 In abstract algebra, the fundamental theorem on homomorphisms, also known as the fundamental homomorphism theorem, or the first isomorphism theorem, relates the structure of two objects between which a homomorphism is given, and of the kernel and ...
and the isomorphism theorems. If ''$G$'' is abelian,
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 cla ...
, solvable, cyclic 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 ta ...
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 In mathematics, a group extension is a general means of describing a group in terms of a particular normal subgroup and quotient group. If Q and N are two groups, then G is an extension of Q by N if there is a short exact sequence :1\to N\;\ove ...
. 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, 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 add ...
and ''$N$'' is a normal and closed (in the topological rather than the algebraic sense of the word) Lie subgroup 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 ...
(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 man ...
. 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 m ...
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 of '' ...
.