In

_{''i''+1} is an extension of ''A''_{''i''} by some

Group Extensions and ''H''^{3}

From his collection of short mathematical notes.

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 ...

, a group extension is a general means of describing 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 ...

in terms of a particular 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), ...

and quotient group
A quotient group or factor group is a mathematical group (mathematics), group obtained by aggregating similar elements of a larger group using an equivalence relation that preserves some of the group structure (the rest of the structure is "factore ...

. If ''Q'' and ''N'' are two groups, then ''G'' is an extension of ''Q'' by ''N'' if there is a 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 ...

:$1\backslash to\; N\backslash ;\backslash overset\backslash ;G\backslash ;\backslash overset\backslash ;Q\; \backslash to\; 1.$
If ''G'' is an extension of ''Q'' by ''N'', then ''G'' is a group, $\backslash iota(N)$ 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 ''G'' and the quotient group
A quotient group or factor group is a mathematical group (mathematics), group obtained by aggregating similar elements of a larger group using an equivalence relation that preserves some of the group structure (the rest of the structure is "factore ...

$G/\backslash iota(N)$ is isomorphic
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 ...

to the group ''Q''. Group extensions arise in the context of the extension problem, where the groups ''Q'' and ''N'' are known and the properties of ''G'' are to be determined. Note that the phrasing "''G'' is an extension of ''N'' by ''Q''" is also used by some.
Since 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) ...

''G'' possesses a maximal 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), ...

''N'' with simple
Simple or SIMPLE may refer to:
*Simplicity, the state or quality of being simple
Arts and entertainment
* Simple (album), ''Simple'' (album), by Andy Yorke, 2008, and its title track
* Simple (Florida Georgia Line song), "Simple" (Florida Ge ...

factor group ''G''/''N'', all finite groups may be constructed as a series of extensions with finite simple 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 and ...

s. This fact was a motivation for completing the classification of finite simple groups
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 ...

.
An extension is called a central extension if the subgroup ''N'' lies 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 spe ...

of ''G''.
Extensions in general

One extension, thedirect product 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 the ...

, is immediately obvious. If one requires ''G'' and ''Q'' to be 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 ...

s, then the set of isomorphism classes of extensions of ''Q'' by a given (abelian) group ''N'' is in fact a group, which is isomorphic
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 ...

to
:$\backslash operatorname^1\_(Q,N);$
cf. the Ext functor In mathematics, the Ext functors are the derived functors of the Hom functor. Along with the Tor functor, Ext is one of the core concepts of homological algebra, in which ideas from algebraic topology are used to define invariants of algebraic struc ...

. Several other general classes of extensions are known but no theory exists that treats all the possible extensions at one time. Group extension is usually described as a hard problem; it is termed the extension problem.
To consider some examples, if , then ''G'' is an extension of both ''H'' and ''K''. More generally, if ''G'' is a semidirect product
In mathematics, specifically in group theory, the concept of a semidirect product is a generalization of a direct product of groups, direct product. There are two closely related concepts of semidirect product:
* an ''inner'' semidirect product ...

of ''K'' and ''H'', written as $G=K\backslash rtimes\; H$, then ''G'' is an extension of ''H'' by ''K'', so such products as the wreath product
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 ( ...

provide further examples of extensions.
Extension problem

The question of what groups ''G'' are extensions of ''H'' by ''N'' is called the extension problem, and has been studied heavily since the late nineteenth century. As to its motivation, consider that thecomposition series 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), r ...

of a finite group is a finite sequence of subgroups , where each ''A''simple 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 and ...

. The classification of finite simple groups
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 ...

gives us a complete list of finite simple groups; so the solution to the extension problem would give us enough information to construct and classify all finite groups in general.
Classifying extensions

Solving the extension problem amounts to classifying all extensions of ''H'' by ''K''; or more practically, by expressing all such extensions in terms of mathematical objects that are easier to understand and compute. In general, this problem is very hard, and all the most useful results classify extensions that satisfy some additional condition. It is important to know when two extensions are equivalent or congruent. We say that the extensions :$1\; \backslash to\; K\backslash stackrel\; G\backslash stackrel\; H\backslash to\; 1$ and :$1\backslash to\; K\backslash stackrel\; G\text{'}\backslash stackrel\; H\backslash to\; 1$ are equivalent (or congruent) if there exists a group isomorphism $T:\; G\backslash to\; G\text{'}$ making commutative the diagram of Figure 1. In fact it is sufficient to have a group homomorphism; due to the assumed commutativity of the diagram, the map $T$ is forced to be an isomorphism by the short five lemma.Warning

It may happen that the extensions $1\backslash to\; K\backslash to\; G\backslash to\; H\backslash to\; 1$ and $1\backslash to\; K\backslash to\; G^\backslash prime\backslash to\; H\backslash to\; 1$ are inequivalent but ''G'' and ''G are isomorphic as groups. For instance, there are $8$ inequivalent extensions of theKlein four-group
In mathematics, the Klein four-group is a Group (mathematics), group with four elements, in which each element is Involution (mathematics), self-inverse (composing it with itself produces the identity)
and in which composing any two of the three ...

by $\backslash mathbb/2\backslash mathbb$, but there are, up to group isomorphism, only four groups of order $8$ containing a normal subgroup of order $2$ with quotient group isomorphic to the Klein four-group
In mathematics, the Klein four-group is a Group (mathematics), group with four elements, in which each element is Involution (mathematics), self-inverse (composing it with itself produces the identity)
and in which composing any two of the three ...

.
Trivial extensions

A trivial extension is an extension :$1\backslash to\; K\backslash to\; G\backslash to\; H\backslash to\; 1$ that is equivalent to the extension :$1\backslash to\; K\backslash to\; K\backslash times\; H\backslash to\; H\backslash to\; 1$ where the left and right arrows are respectively the inclusion and the projection of each factor of $K\backslash times\; H$.Classifying split extensions

A split extension is an extension :$1\backslash to\; K\backslash to\; G\backslash to\; H\backslash to\; 1$ with ahomomorphism
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 ...

$s\backslash colon\; H\; \backslash to\; G$ such that going from ''H'' to ''G'' by ''s'' and then back to ''H'' by the quotient map of the short exact sequence induces the identity map
Graph of the identity function on the real numbers
In mathematics, an identity function, also called an identity relation, identity map or identity transformation, is a function (mathematics), function that always returns the same value that was ...

on ''H'' i.e., $\backslash pi\; \backslash circ\; s=\backslash mathrm\_H$. In this situation, it is usually said that ''s'' splits the above exact sequence
An exact sequence is a sequence of morphisms between objects (for example, groups
A group is a number of people or things that are located, gathered, or classed together.
Groups of people
* Cultural group, a group whose members share the same ...

.
Split extensions are very easy to classify, because an extension is split 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 group ''G'' is a semidirect product
In mathematics, specifically in group theory, the concept of a semidirect product is a generalization of a direct product of groups, direct product. There are two closely related concepts of semidirect product:
* an ''inner'' semidirect product ...

of ''K'' and ''H''. Semidirect products themselves are easy to classify, because they are in one-to-one correspondence with homomorphisms from $H\backslash to\backslash operatorname(K)$, where Aut(''K'') is the 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 ...

group of ''K''. For a full discussion of why this is true, see semidirect product
In mathematics, specifically in group theory, the concept of a semidirect product is a generalization of a direct product of groups, direct product. There are two closely related concepts of semidirect product:
* an ''inner'' semidirect product ...

.
Warning on terminology

In general in mathematics, an extension of a structure ''K'' is usually regarded as a structure ''L'' of which ''K'' is a substructure. See for examplefield 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 t ...

. However, in group theory the opposite terminology has crept in, partly because of the notation $\backslash operatorname(Q,N)$, which reads easily as extensions of ''Q'' by ''N'', and the focus is on the group ''Q''.
A paper of Ronald Brown and Timothy Porter on Otto Schreier
Otto Schreier (3 March 1901 in Vienna
Vienna ( ; german: Wien ; bar, Wean, label=Bavarian language, Austro-Bavarian ) is the Capital city, national capital, largest city, and one of States of Austria, nine states of Austria. Vienna is Austria's ...

's theory of nonabelian extensions uses the terminology that an extension of ''K'' gives a larger structure.
Central extension

A central extension of a group ''G'' is a shortexact sequence
An exact sequence is a sequence of morphisms between objects (for example, groups
A group is a number of people or things that are located, gathered, or classed together.
Groups of people
* Cultural group, a group whose members share the same ...

of groups
:$1\backslash to\; A\backslash to\; E\backslash to\; G\backslash to\; 1$
such that ''A'' is included in $Z(E)$, 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 spe ...

of the group ''E''. The set of isomorphism classes of central extensions of ''G'' by ''A'' (where ''G'' acts trivially on ''A'') is in one-to-one correspondence with the cohomology
In mathematics, specifically in homology theory and algebraic topology, cohomology is a general term for a sequence of abelian groups, usually one associated with a topological space, often defined from a cochain complex. Cohomology can be viewed a ...

group $H^2(G,A)$.
Examples of central extensions can be constructed by taking any group ''G'' and any 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 ...

''A'', and setting ''E'' to be $A\backslash times\; G$. This kind of ''split'' example corresponds to the element 0 in $H^2(G,A)$ under the above correspondence. More serious examples are found in the theory of projective representation In the field of representation theory in mathematics, a projective representation of a group (mathematics), group ''G'' on a vector space ''V'' over a field (mathematics), field ''F'' is a group homomorphism from ''G'' to the projective linear group ...

s, in cases where the projective representation cannot be lifted to an ordinary linear representation
Representation theory is a branch of mathematics that studies abstract algebra, abstract algebraic structures by ''representing'' their element (set theory), elements as linear transformations of vector spaces, and studies Module (mathematics), ...

.
In the case of finite perfect 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 and th ...

s, there is a universal perfect central extension
In mathematical group theory, the Schur multiplier or Schur multiplicator is the second group homology, homology group H_2(G, \Z) of a group ''G''. It was introduced by in his work on projective representations.
Examples and properties
The Schur ...

.
Similarly, the central extension of a Lie algebra
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 ...

$\backslash mathfrak$ is an exact sequence
:$0\backslash rightarrow\; \backslash mathfrak\backslash rightarrow\backslash mathfrak\backslash rightarrow\backslash mathfrak\backslash rightarrow\; 0$
such that $\backslash mathfrak$ is in the center of $\backslash mathfrak$.
There is a general theory of central extensions in Maltsev varieties.
Generalization to general extensions

There is a similar classification of all extensions of ''G'' by ''A'' in terms of homomorphisms from $G\backslash to\backslash operatorname(A)$, a tedious but explicitly checkable existence condition involving and the cohomology group .P. J. MorandiGroup Extensions and ''H''

From his collection of short mathematical notes.

Lie groups

InLie 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 ...

theory, central extensions arise in connection with algebraic topology
Algebraic topology is a branch of 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 ...

. Roughly speaking, central extensions of Lie groups by discrete groups are the same as covering group
In mathematics, a covering group of a topological group ''H'' is a covering space ''G'' of ''H'' such that ''G'' is a topological group and the covering map is a continuous (topology), continuous group homomorphism. The map ''p'' is called the c ...

s. More precisely, a connected covering space
In mathematics, specifically algebraic topology, a covering map (also covering projection) is a continuous function p from a topological space C to a topological space
X such that each point in X has an Neighborhood_(mathematics)#Definitions, ...

of a connected Lie group is naturally a central extension of , in such a way that the projection
:$\backslash pi\backslash colon\; G^*\; \backslash to\; G$
is a group homomorphism, and surjective. (The group structure on depends on the choice of an identity element mapping to the identity in .) For example, when is the universal cover
In mathematics, specifically algebraic topology, a covering map (also covering projection) is a continuous function p from a topological space C to a topological space
X such that each point in X has an Neighborhood_(mathematics)#Definitions, ...

of , the kernel of ''π'' is the fundamental group
In the mathematical
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 ...

of , which is known to be abelian (see H-space In mathematics, an H-space is a homotopy theory, homotopy-theoretic version of a generalization of the notion of topological group, in which the axioms on associativity and inverse element, inverses are removed.
Definition
An H-space consists of a ...

). Conversely, given a Lie group and a discrete central subgroup , the quotient is a Lie group and is a covering space of it.
More generally, when the groups , and occurring in a central extension are Lie groups, and the maps between them are homomorphisms of Lie groups, then if the Lie algebra of is , that of is , and that of is , then is a central Lie algebra extension of by . In the terminology of theoretical physics
Theoretical physics is a branch of physics that employs mathematical models and abstractions of physical objects and systems to rationalize, explain and predict List of natural phenomena, natural phenomena. This is in contrast to experimental ph ...

, generators of are called central charge
In theoretical physics, a central charge is an operator (mathematics), operator ''Z'' that Commutativity, commutes with all the other symmetry operators. The adjective "central" refers to the center (group theory), center of the symmetry group—th ...

s. These generators are in the center of ; by Noether's theorem
Noether's theorem or Noether's first theorem states that every differentiable
In calculus (a branch of mathematics), a differentiable function of one Real number, real variable is a function whose derivative exists at each point in its Domai ...

, generators of symmetry groups correspond to conserved quantities, referred to as charges.
The basic examples of central extensions as covering groups are:
* the spin group In mathematics the spin group Spin(''n'') page 15 is the covering space, double cover of the special orthogonal group , such that there exists a short exact sequence of Lie groups (when )
:1 \to \mathrm_2 \to \operatorname(n) \to \operatorname(n) \ ...

s, which double cover the special orthogonal 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 and ...

s, which (in even dimension) doubly cover the projective orthogonal group In projective geometry
In mathematics, projective geometry is the study of geometric properties that are invariant with respect to projective transformations. This means that, compared to elementary Euclidean geometry, projective geometry has a ...

.
* the metaplectic groups, which double cover the symplectic group
In mathematics, the name symplectic group can refer to two different, but closely related, collections of mathematical Group (mathematics), groups, denoted and for positive integer ''n'' and field (mathematics), field F (usually C or R). The la ...

s.
The case of involves a fundamental group that is infinite cyclic
In group theory, a branch of abstract algebra, a cyclic group or monogenous group is a Group (mathematics), group that is Generating set of a group, generated by a single element. That is, it is a set (mathematics), set of Inverse element, inverti ...

. Here the central extension involved is well known in modular form
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 ...

theory, in the case of forms of weight . A projective representation that corresponds is the Weil representation, constructed from the Fourier transform#REDIRECT Fourier transform
In mathematics, a Fourier transform (FT) is a Integral transform, mathematical transform that decomposes function (mathematics), functions depending on space or time into functions depending on spatial or temporal frequenc ...

, in this case on the real line
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 ...

. Metaplectic groups also occur in quantum mechanics
Quantum mechanics is a fundamental theory
A theory is a reason, rational type of abstraction, abstract thinking about a phenomenon, or the results of such thinking. The process of contemplative and rational thinking is often associated with ...

.
See also

*Lie algebra extension
In the theory of Lie groups, Lie algebras and their Lie algebra representation, representation theory, a Lie algebra extension is an enlargement of a given Lie algebra by another Lie algebra . Extensions arise in several ways. There is the trivi ...

*Ring extension
In commutative algebra, a ring extension is a ring homomorphism R\to S of commutative rings, which makes an -algebra over a ring, algebra.
In this article, a ring extension of a ring (mathematics), ring ''R'' by an abelian group ''I'' is a pair o ...

*Virasoro algebra
In mathematics, the Virasoro algebra (named after the physicist Miguel Ángel Virasoro (physicist), Miguel Ángel Virasoro) is a complex Lie algebra and the unique Lie algebra extension#Central, central extension of the Witt algebra. It is widely ...

*HNN extension In mathematics, the HNN extension is an important construction of combinatorial group theory.
Introduced in a 1949 paper ''Embedding Theorems for Groups'' by Graham Higman, Bernhard Neumann, and Hanna Neumann, it embeds a given group ''G'' into ano ...

*Group contraction
In theoretical physics, Eugene Wigner
Eugene Paul "E. P." Wigner ( hu, Wigner Jenő Pál, ; November 17, 1902 – January 1, 1995) was a Hungarian-American theoretical physicist and also contributed to mathematical physics. He received the Nobel ...

* Extension of a topological group
References

*{{Citation , first=Saunders , last1=Mac Lane , authorlink = Saunders Mac Lane, title=Homology , publisher=Springer Verlag
Springer Science+Business Media, commonly known as Springer, is a German multinational publishing
Publishing is the activity of making information, literature, music, software and other content available to the public for sale or for free. ...

, year=1975 , isbn=3-540-58662-8 , series=Classics in Mathematics
* R.L. Taylor, Covering groups of non connected topological groups, ''Proceedings of the American Mathematical Society
In academia and librarianship, conference proceeding is a collection of academic paper
Academic publishing is the subfield of publishing which distributes academic research and scholarship. Most academic work is published in academic journal a ...

'', vol. 5 (1954), 753–768.
* R. Brown and O. Mucuk, Covering groups of non-connected topological groups revisited, ''Mathematical Proceedings of the Cambridge Philosophical Society
{{wikisource, Proceedings of the Cambridge Philosophical Society
''Mathematical Proceedings of the Cambridge Philosophical Society'' is a mathematical journal published by Cambridge University Press for the Cambridge Philosophical Society. It aims ...

'', vol. 115 (1994), 97–110.
Group theory