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A quotient group or factor group is a
math 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 ... ematical
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 ...
obtained by aggregating similar elements of a larger group using an
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). ...
that preserves some of the group structure (the rest of the structure is "factored" out). For example, the
cyclic group In group theory The popular puzzle Rubik's cube invented in 1974 by Ernő Rubik has been used as an illustration of permutation group">Ernő_Rubik.html" ;"title="Rubik's cube invented in 1974 by Ernő Rubik">Rubik's cube invented in 1974 by Er ... of addition modulo ''n'' can be obtained from the group of
integer An integer (from the Latin Latin (, or , ) is a classical language A classical language is a language A language is a structured system of communication Communication (from Latin ''communicare'', meaning "to share" or "to ...
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 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 a single entity. It is part of the mathematical field known as
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 ...
. In a quotient of a group, the
equivalence 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). ...
of the
identity element 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 ...
is always 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 the original group, and the other equivalence classes are precisely the
coset 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 of that normal subgroup. The resulting quotient is written , where ''G'' is the original group and ''N'' is the normal subgroup. (This is pronounced "''G'' mod ''N''", where "mod" 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 for ... states that the
image An image (from la, imago) is an artifact that depicts visual perception Visual perception is the ability to interpret the surrounding environment (biophysical), environment through photopic vision (daytime vision), color vision, sco ...
of any group ''G'' under a homomorphism is always
isomorphic 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 ...
to a quotient of ''G''. Specifically, the image of ''G'' under a homomorphism is isomorphic to where ker(''φ'') 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 learnin ...
of ''φ''. The
dual Dual or Duals may refer to: Paired/two things * Dual (mathematics), a notion of paired concepts that mirror one another ** Dual (category theory), a formalization of mathematical duality ** . . . see more cases in :Duality theories * Dual ...
notion of a quotient group is 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 ...
, 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 formalizes mathematical structure In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and ...
, quotient groups are examples of
quotient objectIn category theory Category theory formalizes mathematical structure and its concepts in terms of a Graph labeling, labeled directed graph called a ''Category (mathematics), category'', whose nodes are called ''objects'', and whose labelled dire ...
s, which are
dual Dual or Duals may refer to: Paired/two things * Dual (mathematics), a notion of paired concepts that mirror one another ** Dual (category theory), a formalization of mathematical duality ** . . . see more cases in :Duality theories * Dual ...
to
subobject In category theory Category theory formalizes mathematical structure In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), sp ...
s.

# Definition and illustration

Given 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 ...
''G'' and a subgroup ''H'', and an element ''a'' ∈ ''G'', one can consider the corresponding left
coset 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). ...
: ''aH'' := . Cosets are a natural class of subsets of a group; for example consider the
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 ...
''G'' of
integer An integer (from the Latin Latin (, or , ) is a classical language A classical language is a language A language is a structured system of communication Communication (from Latin ''communicare'', meaning "to share" or "to ...
s, with operation 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, . 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 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 ...
the coset equality ''aN'' = ''Na'' holds for all ''a'' ∈ ''G''. A normal subgroup of ''G'' is denoted .

## 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, . Since the identity element ''e'' ∈ ''N'', ''a'' ∈ ''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'' ∈ ''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'' ∈ ''G'', (''ab'')''N'' = (''xy'')''N.'' Let ''n'' ∈ ''N'' and ''g'' ∈ ''G''. Since ''eN'' = ''nN,'' we have, ''gN'' = (''eg'')''N'' = (''ng'')''N.'' Now, ''gN'' = (''ng'')''N'' ⇔ ''N'' = ''g''−1(''ng'')''N'' ⇔ ''g''−1''ng'' ∈ ''N'' ∀ ''n'' ∈ ''N'' and ''g'' ∈ ''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''−1''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''.

For example, consider the group with addition modulo 6: ''G'' = . Consider the subgroup ''N'' = , which is normal because ''G'' is abelian. Then the set of (left) cosets is of size three: : ''G''/''N'' = = = . 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 In group theory The popular puzzle Rubik's cube invented in 1974 by Ernő Rubik has been used as an illustration of permutation group">Ernő_Rubik.html" ;"title="Rubik's cube invented in 1974 by Ernő Rubik">Rubik's cube invented in 1974 by Er ... of order 3.

# Motivation for the name "quotient"

The reason ''G''/''N'' is called a quotient group comes from
division Division or divider may refer to: Mathematics *Division (mathematics), the inverse of multiplication *Division algorithm, a method for computing the result of mathematical division Military *Division (military), a formation typically consisting o ...
of
integer An integer (from the Latin Latin (, or , ) is a classical language A classical language is a language A language is a structured system of communication Communication (from Latin ''communicare'', meaning "to share" or "to ...
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 (from the Latin Latin (, or , ) is a classical language A classical language is a language A language is a structured system of communication Communication (from Latin ''communicare'', meaning "to share" or "to ...
s Z (under addition) and the subgroup 2Z 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/2Z is the cyclic group with two elements. This quotient group is isomorphic with the set with addition modulo 2; informally, it is sometimes said that Z/2Z ''equals'' the set with addition modulo 2. Example further explained... : Let $\gamma\left(m\right)=$ 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($\gamma$) $= \$, 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 $\mu\left(aH\right)=\gamma\left(a\right)$ for $a\in\Z$ : and is the quotient group of left cosets; $=\$. : By the way 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 . 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 In group theory The popular puzzle Rubik's cube invented in 1974 by Ernő Rubik has been used as an illustration of permutation group">Ernő_Rubik.html" ;"title="Rubik's cube invented in 1974 by Ernő Rubik">Rubik's cube invented in 1974 by Er ... of order ''n''.

## Complex integer roots of 1 The twelfth
roots of unity 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) ...
, which are points on the
complex The UCL Faculty of Mathematical and Physical Sciences is one of the 11 constituent faculties of University College London , mottoeng = Let all come who by merit deserve the most reward , established = , type = Public university, Public rese ... unit circle 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 a ... , 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 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 g ...
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 ''a1''+Z and ''a2''+Z are identical sets when the non-
integer part 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 of ''a1'' and ''a2'' are equal, one may impose the restriction 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 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 g ...
, the group of
complex number In mathematics, a complex number is an element of a number system that contains the real numbers and a specific element denoted , called the imaginary unit, and satisfying the equation . Moreover, every complex number can be expressed in the for ... s of
absolute value 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 ... 1 under multiplication, or correspondingly, the group of
rotation A rotation is a circular movement of an object around a center (or point) of rotation. The plane (geometry), geometric plane along which the rotation occurs is called the ''rotation plane'', and the imaginary line extending from the center an ... s in 2D about the origin, that is, the special
orthogonal group 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 ...
SO(2). An isomorphism is given by (see
Euler's identity In mathematics, Euler's identity (also known as Euler's equation) is the Equality (mathematics), equality :e^ + 1 = 0 where : is E (mathematical constant), Euler's number, the base of natural logarithms, : is the imaginary unit, which by defini ...
).

## Matrices of real numbers

If ''G'' is the group of invertible 3 × 3 real
matrices Matrix or MATRIX may refer to: Science and mathematics * Matrix (mathematics) In mathematics, a matrix (plural matrices) is a rectangle, rectangular ''wikt:array, array'' or ''table'' of numbers, symbol (formal), symbols, or expression (mathema ...
, and ''N'' is the subgroup of 3 × 3 real matrices with
determinant 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 t ... 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 learnin ...
of the determinant
homomorphism In algebra Algebra (from ar, الجبر, lit=reunion of broken parts, bonesetting, translit=al-jabr) is one of the areas of mathematics, broad areas of mathematics, together with number theory, geometry and mathematical analysis, analysis. I ... ). 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 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 ...
SL(3).

## Integer modular arithmetic

Consider the abelian group (that is, the set with addition modulo 4), and its subgroup . The quotient group is . This is a group with identity element , and group operations such as . Both the subgroup and the quotient group are isomorphic with Z2.

## Integer multiplication

Consider the multiplicative group $G=\mathbf^*_$. The set ''N'' of ''n''th residues is a multiplicative subgroup isomorphic to $\mathbf^*_$. Then ''N'' is normal in ''G'' and the factor group ''G''/''N'' has the cosets ''N'', (1+''n'')''N'', (1+''n'')2N, ..., (1+''n'')''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 File:Private key signing.svg, 250px, In this example the message is digital signature, digital ...
is based on the
conjecture 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 ...
that it is difficult to determine the coset of a random element of ''G'' without knowing the factorization of ''n''.

# Properties

The quotient group is
isomorphic 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 ...
to the
trivial groupIn 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 ha ...
(the group with one element), and is isomorphic to ''G''. The
order Order, ORDER or Orders may refer to: * Orderliness Orderliness is a quality that is characterized by a person’s interest in keeping their surroundings and themselves well organized, and is associated with other qualities such as cleanliness a ...
of , by definition the number of elements, is equal to , 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 ''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 may be finite, although both ''G'' and ''N'' are infinite (for example, ). There is a "natural"
surjective 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 ...
group homomorphism 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 ge ... , sending each element ''g'' of ''G'' to the coset of ''N'' to which ''g'' belongs, that is: . The mapping ''π'' is sometimes called the ''canonical projection of G onto ''. 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 learnin ...
is ''N''. There is a bijective correspondence between the subgroups of ''G'' that contain ''N'' and the subgroups of ; if ''H'' is a subgroup of ''G'' containing ''N'', then the corresponding subgroup of is ''π''(''H''). This correspondence holds for normal subgroups of ''G'' and as well, and is formalized in the
lattice theoremIn the area of 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, anal ...
. 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, relates the structure of two objects between which a homomorphism is given, and of the Kernel (algebra), kernel and image of the hom ...
and the
isomorphism theorem 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 ...
s. If ''G'' is abelian,
nilpotent 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 ...
, solvable,
cyclic Cycle 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 social scienc ... or finitely generated, then so is . 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 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 has order ''p'', ''H'' must be a normal subgroup of ''G''. Given ''G'' and a normal subgroup ''N'', then ''G'' is a
group extension 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 ...
of by ''N''. One could ask whether this extension is trivial or split; in other words, one could ask whether ''G'' is a
direct productIn 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 ha ...
or
semidirect product 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 ''N'' and . This is a special case of the extension problem. An example where the extension is not split is as follows: Let ''G'' = Z4 = , and ''N'' = , which is isomorphic to Z2. Then is also isomorphic to Z2. But Z2 has only the trivial
automorphism 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 t ... , so the only semi-direct product of ''N'' and is the direct product. Since Z4 is different from , we conclude that ''G'' is not a semi-direct product of ''N'' and .

# Quotients of Lie groups

If ''$G$'' is a
Lie 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 a ...
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 English in the Commonwealth of Nations, Commonwealth English: fibre bundle) is a Space (mathematics), space that is ''locally'' a product space, but ''globally'' may have a dif ...
(specifically, a principal ''$N$''-bundle), with base space and fiber ''$N$''. The dimension of equals $\mathrm\ G - \mathrm\ N$.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 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), ... . For a non-normal Lie subgroup ''$N$'', the space 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 The real projective plane is a two-dimensional manifold that cannot be realized in three dimensions without self-intersection, shown here as Boy's surfa ...
on which ''$G$'' acts. The result is known as a
homogeneous 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). It ...
.

*
Group extension 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 ...
*
Quotient category In mathematics, a quotient category is a category (mathematics), category obtained from another one by identifying sets of morphisms. Formally, it is a quotient object in the category of small categories, category of (locally small) categories, anal ...
*
Short exact sequence An exact sequence is a sequence of morphisms between objects (for example, Group (mathematics), groups, Ring (mathematics), rings, Module (mathematics), modules, and, more generally, objects of an abelian category) such that the Image (mathematics ...

# 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 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 ...