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In
mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and their changes (cal ...
, 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\to N\;\overset\;G\;\overset\;Q \to 1. If ''G'' is an extension of ''Q'' by ''N'', then ''G'' is a group, \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/\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 ...

isomorphic
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, the
direct 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 ...

isomorphic
to :\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\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 the
composition 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''''i''+1 is an extension of ''A''''i'' by some
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 \to K\stackrel G\stackrel H\to 1 and :1\to K\stackrel G'\stackrel H\to 1 are equivalent (or congruent) if there exists a group isomorphism T: G\to G' 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\to K\to G\to H\to 1 and 1\to K\to G^\prime\to H\to 1 are inequivalent but ''G'' and ''G are isomorphic as groups. For instance, there are 8 inequivalent extensions of 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 ...
by \mathbb/2\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\to K\to G\to H\to 1 that is equivalent to the extension :1\to K\to K\times H\to H\to 1 where the left and right arrows are respectively the inclusion and the projection of each factor of K\times H.


Classifying split extensions

A split extension is an extension :1\to K\to G\to H\to 1 with a
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 ...
s\colon H \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 ...

identity map
on ''H'' i.e., \pi \circ s=\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\to\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 ...

automorphism
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 example
field 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 \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 short
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 ...
of groups :1\to A\to E\to G\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\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 ...
\mathfrak is an exact sequence :0\rightarrow \mathfrak\rightarrow\mathfrak\rightarrow\mathfrak\rightarrow 0 such that \mathfrak is in the center of \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\to\operatorname(A), a tedious but explicitly checkable existence condition involving and the cohomology group .P. J. Morandi
Group Extensions and ''H''3
From his collection of short mathematical notes.


Lie groups

In
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 ...
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 :\pi\colon G^* \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 ...

fundamental group
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