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In
mathematics 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 ...
, a group extension is a general means of describing a
group A group is a number of persons or things that are located, gathered, or classed together. Groups of people * Cultural group, a group whose members share the same cultural identity * Ethnic group, a group whose members share the same ethnic ide ...
in terms of a particular
normal subgroup In abstract algebra, a normal subgroup (also known as an invariant subgroup or self-conjugate subgroup) is a subgroup that is invariant under conjugation by members of the group of which it is a part. In other words, a subgroup N of the group G ...
and
quotient group A quotient group or factor group is a mathematical 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 "factored" out). For exam ...
. 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, groups, rings, modules, and, more generally, objects of an abelian category) such that the image of one morphism equals the kernel of the next. Definition In the context ...
: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, a normal subgroup (also known as an invariant subgroup or self-conjugate subgroup) is a subgroup that is invariant under conjugation by members of the group of which it is a part. In other words, a subgroup N of the group G ...
of G and the
quotient group A quotient group or factor group is a mathematical 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 "factored" out). For exam ...
G/\iota(N) 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 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 Finite is the opposite of infinite. It may refer to: * Finite number (disambiguation) * Finite set, a set whose cardinality (number of elements) is some natural number * Finite verb, a verb form that has a subject, usually being inflected or ma ...
G possesses a maximal
normal subgroup In abstract algebra, a normal subgroup (also known as an invariant subgroup or self-conjugate subgroup) is a subgroup that is invariant under conjugation by members of the group of which it is a part. In other words, a subgroup N of the group G ...
N with
simple Simple or SIMPLE may refer to: *Simplicity, the state or quality of being simple Arts and entertainment * ''Simple'' (album), by Andy Yorke, 2008, and its title track * "Simple" (Florida Georgia Line song), 2018 * "Simple", a song by Johnn ...
factor group G/N, all finite groups may be constructed as a series of extensions with finite
simple group SIMPLE Group Limited is a conglomeration of separately run companies that each has its core area in International Consulting. The core business areas are Legal Services, Fiduciary Activities, Banking Intermediation and Corporate Service. The d ...
s. This fact was a motivation for completing the
classification of finite simple groups In mathematics, the classification of the finite simple groups is a result of group theory stating that every finite simple group is either cyclic, or alternating, or it belongs to a broad infinite class called the groups of Lie type, or else i ...
. An extension is called a central extension if the subgroup N lies in the
center Center or centre may refer to: Mathematics *Center (geometry), the middle of an object * Center (algebra), used in various contexts ** Center (group theory) ** Center (ring theory) * Graph center, the set of all vertices of minimum eccentrici ...
of G.


Extensions in general

One extension, the
direct product In mathematics, one can often define a direct product of objects already known, giving a new one. This generalizes the Cartesian product of the underlying sets, together with a suitably defined structure on the product set. More abstractly, one t ...
, is immediately obvious. If one requires G and Q to be
abelian group In mathematics, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on the order in which they are written. That is, the group operation is comm ...
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, 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 :\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 stru ...
. 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. There are two closely related concepts of semidirect product: * an ''inner'' semidirect product is a particular way in wh ...
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, the wreath product is a special combination of two groups based on the semidirect product. It is formed by the action of one group on many copies of another group, somewhat analogous to exponentiation. Wreath products are used i ...
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 of a finite group is a finite sequence of subgroups \, where each \ is an extension of \ by some
simple group SIMPLE Group Limited is a conglomeration of separately run companies that each has its core area in International Consulting. The core business areas are Legal Services, Fiduciary Activities, Banking Intermediation and Corporate Service. The d ...
. The
classification of finite simple groups In mathematics, the classification of the finite simple groups is a result of group theory stating that every finite simple group is either cyclic, or alternating, or it belongs to a broad infinite class called the groups of Lie type, or else i ...
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 In mathematics, especially homological algebra and other applications of abelian category theory, the short five lemma is a special case of the five lemma. It states that for the following commutative diagram (in any abelian category, or in the c ...
.


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 with four elements, in which each element is self-inverse (composing it with itself produces the identity) and in which composing any two of the three non-identity elements produces the third one ...
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 with four elements, in which each element is self-inverse (composing it with itself produces the identity) and in which composing any two of the three non-identity elements produces the third one ...
.


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, a homomorphism is a morphism, structure-preserving map (mathematics), map between two algebraic structures of the same type (such as two group (mathematics), groups, two ring (mathematics), rings, or two vector spaces). The word ''homo ...
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 that always returns the value that was used as its argument, unc ...
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, rings, modules, and, more generally, objects of an abelian category) such that the image of one morphism equals the kernel of the next. Definition In the context ...
. Split extensions are very easy to classify, because an extension is split
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 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. There are two closely related concepts of semidirect product: * an ''inner'' semidirect product is a particular way in wh ...
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, an automorphism is an isomorphism from a mathematical object to itself. It is, in some sense, a symmetry of the object, and a way of mapping the object to itself while preserving all of its structure. The set of all automorphis ...
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. There are two closely related concepts of semidirect product: * an ''inner'' semidirect product is a particular way in wh ...
.


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, particularly in algebra, a field extension is a pair of fields E\subseteq F, such that the operations of ''E'' are those of ''F'' restricted to ''E''. In this case, ''F'' is an extension field of ''E'' and ''E'' is a subfield of ...
. 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'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, rings, modules, and, more generally, objects of an abelian category) such that the image of one morphism equals the kernel of the next. Definition In the context ...
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), the middle of an object * Center (algebra), used in various contexts ** Center (group theory) ** Center (ring theory) * Graph center, the set of all vertices of minimum eccentrici ...
of the group ''E''. The set of isomorphism classes of central extensions of ''G'' by ''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 view ...
group H^2(G,A). Examples of central extensions can be constructed by taking any group ''G'' and any
abelian group In mathematics, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on the order in which they are written. That is, the group operation is comm ...
''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 ''G'' on a vector space ''V'' over a field ''F'' is a group homomorphism from ''G'' to the projective linear group \mathrm(V) = \mathrm(V) / F^*, where ...
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 algebraic structures by ''representing'' their elements as linear transformations of vector spaces, and studies modules over these abstract algebraic structures. In essenc ...
. In the case of finite
perfect group In mathematics, more specifically in group theory, a group is said to be perfect if it equals its own commutator subgroup, or equivalently, if the group has no non-trivial abelian quotients (equivalently, its abelianization, which is the unive ...
s, there is a
universal perfect central extension In mathematical group theory, the Schur multiplier or Schur multiplicator is the second 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 multiplier \oper ...
. Similarly, the central extension of a
Lie algebra In mathematics, a Lie algebra (pronounced ) is a vector space \mathfrak g together with an operation called the Lie bracket, an alternating bilinear map \mathfrak g \times \mathfrak g \rightarrow \mathfrak g, that satisfies the Jacobi identi ...
\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, a Lie group (pronounced ) is a group that is also a differentiable manifold. A manifold is a space that locally resembles Euclidean space, whereas groups define the abstract concept of a binary operation along with the addi ...
theory, central extensions arise in connection with
algebraic topology Algebraic topology is a branch of mathematics that uses tools from abstract algebra to study topological spaces. The basic goal is to find algebraic invariants that classify topological spaces up to homeomorphism, though usually most classify ...
. 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 group homomorphism. The map ''p'' is called the covering homomorphism. ...
s. More precisely, a
connected Connected may refer to: Film and television * ''Connected'' (2008 film), a Hong Kong remake of the American movie ''Cellular'' * '' Connected: An Autoblogography About Love, Death & Technology'', a 2011 documentary film * ''Connected'' (2015 TV ...
covering space A covering of a topological space X is a continuous map \pi : E \rightarrow X with special properties. Definition Let X be a topological space. A covering of X is a continuous map : \pi : E \rightarrow X such that there exists a discrete spa ...
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 A covering of a topological space X is a continuous map \pi : E \rightarrow X with special properties. Definition Let X be a topological space. A covering of X is a continuous map : \pi : E \rightarrow X such that there exists a discrete spa ...
of , the kernel of ''π'' is the
fundamental group In the mathematical field of algebraic topology, the fundamental group of a topological space is the group of the equivalence classes under homotopy of the loops contained in the space. It records information about the basic shape, or holes, o ...
of , which is known to be abelian (see
H-space In mathematics, an H-space is a homotopy-theoretic version of a generalization of the notion of topological group, in which the axioms on associativity and inverses are removed. Definition An H-space consists of a topological space , together w ...
). 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 natural phenomena. This is in contrast to experimental physics, which uses experim ...
, generators of are called
central charge In theoretical physics, a central charge is an operator ''Z'' that commutes with all the other symmetry operators. The adjective "central" refers to the center of the symmetry group—the subgroup of elements that commute with all other elemen ...
s. These generators are in the center of ; by
Noether's theorem Noether's theorem or Noether's first theorem states that every differentiable symmetry of the action of a physical system with conservative forces has a corresponding conservation law. The theorem was proven by mathematician Emmy Noether ...
, generators of symmetry groups correspond to conserved quantities, referred to as
charges Charge or charged may refer to: Arts, entertainment, and media Films * ''Charge, Zero Emissions/Maximum Speed'', a 2011 documentary Music * ''Charge'' (David Ford album) * ''Charge'' (Machel Montano album) * '' Charge!!'', an album by The Aqu ...
. The basic examples of central extensions as covering groups are: * the
spin group In mathematics the spin group Spin(''n'') page 15 is the 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) \to 1. As a ...
s, which double cover the
special orthogonal group In mathematics, the orthogonal group in dimension , denoted , is the group of distance-preserving transformations of a Euclidean space of dimension that preserve a fixed point, where the group operation is given by composing transformations. ...
s, which (in even dimension) doubly cover the
projective orthogonal group In projective geometry and linear algebra, the projective orthogonal group PO is the induced action of the orthogonal group of a quadratic space ''V'' = (''V'',''Q'')A quadratic space is a vector space ''V'' together with a quadratic form ''Q'' ...
. * the
metaplectic group In mathematics, the metaplectic group Mp2''n'' is a double cover of the symplectic group Sp2''n''. It can be defined over either real or ''p''-adic numbers. The construction covers more generally the case of an arbitrary local or finite field ...
s, which double cover the symplectic groups. The case of involves a fundamental group that is infinite cyclic. Here the central extension involved is well known in
modular form In mathematics, a modular form is a (complex) analytic function on the upper half-plane satisfying a certain kind of functional equation with respect to the group action of the modular group, and also satisfying a growth condition. The theory o ...
theory, in the case of forms of weight . A projective representation that corresponds is the Weil representation, constructed from the
Fourier transform A Fourier transform (FT) is a mathematical transform that decomposes functions into frequency components, which are represented by the output of the transform as a function of frequency. Most commonly functions of time or space are transformed ...
, in this case on the
real line In elementary mathematics, a number line is a picture of a graduated straight line that serves as visual representation of the real numbers. Every point of a number line is assumed to correspond to a real number, and every real number to a po ...
. Metaplectic groups also occur in
quantum mechanics Quantum mechanics is a fundamental theory in physics that provides a description of the physical properties of nature at the scale of atoms and subatomic particles. It is the foundation of all quantum physics including quantum chemistry, ...
.


See also

*
Lie algebra extension In the theory of Lie groups, Lie algebras and their 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 trivial extension obtained by ...
*
Ring extension In commutative algebra, a ring extension is a ring homomorphism R\to S of commutative rings, which makes an -algebra. In this article, a ring extension of a ring ''R'' by an abelian group ''I'' is a pair of a ring ''E'' and a surjective ring ho ...
* Virasoro algebra * HNN extension *
Group contraction In theoretical physics, Eugene Wigner and Erdal İnönü have discussed the possibility to obtain from a given Lie group a different (non-isomorphic) Lie group by a group contraction with respect to a continuous subgroup of it. That amounts to a l ...
* 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 company of books, e-books and peer-reviewed journals in science, humanities, technical and medical (STM) publishing. Originally founded in 1842 i ...
, 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 ''Proceedings of the American Mathematical Society'' is a monthly peer-reviewed scientific journal of mathematics published by the American Mathematical Society. As a requirement, all articles must be at most 15 printed pages. According to the ' ...
'', 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 ''Mathematical Proceedings of the Cambridge Philosophical Society'' is a mathematical journal published by Cambridge University Press for the Cambridge Philosophical Society. It aims to publish original research papers from a wide range of pure ...
'', vol. 115 (1994), 97–110. Group theory