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In mathematical
group theory In abstract algebra, group theory studies the algebraic structures known as group (mathematics), groups. The concept of a group is central to abstract algebra: other well-known algebraic structures, such as ring (mathematics), rings, field ...
, the Schur multiplier or Schur multiplicator is the second
homology group In mathematics, homology is a general way of associating a sequence of algebraic objects, such as abelian groups or modules, with other mathematical objects such as topological spaces. Homology groups were originally defined in algebraic topolog ...
H_2(G, \Z) of a group ''G''. It was introduced by in his work on
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 G ...
s.


Examples and properties

The Schur multiplier \operatorname(G) of a finite group ''G'' is a finite
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 commut ...
whose
exponent Exponentiation is a mathematical operation, written as , involving two numbers, the '' base'' and the ''exponent'' or ''power'' , and pronounced as " (raised) to the (power of) ". When is a positive integer, exponentiation corresponds to re ...
divides the order of ''G''. If a Sylow ''p''-subgroup of ''G'' is cyclic for some ''p'', then the order of \operatorname(G) is not divisible by ''p''. In particular, if all Sylow ''p''-subgroups of ''G'' are cyclic, then \operatorname(G) is trivial. For instance, the Schur multiplier of the
nonabelian group of order 6 In mathematics, D3 (sometimes alternatively denoted by D6) is the dihedral group of degree 3, or, in other words, the dihedral group of Order of a group, order 6. It is isomorphic to the symmetric group S3 of degree 3. It is also the smallest ...
is the
trivial group In mathematics, a trivial group or zero group is a group consisting of a single element. All such groups are isomorphic, so one often speaks of the trivial group. The single element of the trivial group is the identity element and so it is usually ...
since every Sylow subgroup is cyclic. The Schur multiplier of the
elementary abelian group In mathematics, specifically in group theory, an elementary abelian group (or elementary abelian ''p''-group) is an abelian group in which every nontrivial element has order ''p''. The number ''p'' must be prime, and the elementary abelian group ...
of order 16 is an elementary abelian group of order 64, showing that the multiplier can be strictly larger than the group itself. The Schur multiplier of the
quaternion group In group theory, the quaternion group Q8 (sometimes just denoted by Q) is a non-abelian group of order eight, isomorphic to the eight-element subset \ of the quaternions under multiplication. It is given by the group presentation :\mathrm_8 ...
is trivial, but the Schur multiplier of dihedral 2-groups has order 2. The Schur multipliers of the 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 are given at the
list of finite simple groups A ''list'' is any set of items in a row. List or lists may also refer to: People * List (surname) Organizations * List College, an undergraduate division of the Jewish Theological Seminary of America * SC Germania List, German rugby union ...
. The
covering groups of the alternating and symmetric groups In the mathematical area of group theory, the covering groups of the alternating and symmetric groups are groups that are used to understand the projective representations of the alternating and symmetric groups. The covering groups were classifie ...
are of considerable recent interest.


Relation to projective representations

Schur's original motivation for studying the multiplier was to classify
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 G ...
s of a group, and the modern formulation of his definition is the second
cohomology group 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 viewe ...
H^2(G, \Complex^). A projective representation is much like a
group representation In the mathematical field of representation theory, group representations describe abstract groups in terms of bijective linear transformations of a vector space to itself (i.e. vector space automorphisms); in particular, they can be used to re ...
except that instead of a homomorphism into the
general linear group In mathematics, the general linear group of degree ''n'' is the set of invertible matrices, together with the operation of ordinary matrix multiplication. This forms a group, because the product of two invertible matrices is again invertible, ...
\operatorname(n, \Complex), one takes a homomorphism into the
projective general linear group In mathematics, especially in the group theoretic area of algebra, the projective linear group (also known as the projective general linear group or PGL) is the induced action of the general linear group of a vector space ''V'' on the associate ...
\operatorname(n, \Complex). In other words, a projective representation is a representation modulo 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 ...
. showed that every finite group ''G'' has associated to it at least one finite group ''C'', called a Schur cover, with the property that every projective representation of ''G'' can be lifted to an ordinary representation of ''C''. The Schur cover is also known as a covering group or Darstellungsgruppe. The Schur covers of the
finite simple groups 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 marked ...
are known, and each is an example of a
quasisimple group In mathematics, a quasisimple group (also known as a covering group) is a group that is a perfect central extension ''E'' of a simple group ''S''. In other words, there is a short exact sequence :1 \to Z(E) \to E \to S \to 1 such that E = , E/ ...
. The Schur cover of a
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 universa ...
is uniquely determined up to isomorphism, but the Schur cover of a general finite group is only determined up to
isoclinism In mathematics, specifically group theory, isoclinism is an equivalence relation on groups which generalizes isomorphism. Isoclinism was introduced by to help classify and understand p-groups, although it is applicable to all groups. Isoclinism al ...
.


Relation to central extensions

The study of such covering groups led naturally to the study of
central Central is an adjective usually referring to being in the center of some place or (mathematical) object. Central may also refer to: Directions and generalised locations * Central Africa, a region in the centre of Africa continent, also known as ...
and stem extensions. A central extension of a group ''G'' is an extension :1 \to K\to C\to G\to 1 where K\le Z(C) is a
subgroup In group theory, a branch of mathematics, given a group ''G'' under a binary operation ∗, a subset ''H'' of ''G'' is called a subgroup of ''G'' if ''H'' also forms a group under the operation ∗. More precisely, ''H'' is a subgroup ...
of 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 ''C''. A stem extension of a group ''G'' is an extension :1 \to K\to C\to G\to 1 where K\le Z(C)\cap C' is a subgroup of the intersection of the center of ''C'' and the
derived subgroup In mathematics, more specifically in abstract algebra, the commutator subgroup or derived subgroup of a group is the subgroup generated by all the commutators of the group. The commutator subgroup is important because it is the smallest normal ...
of ''C''; this is more restrictive than central. If the group ''G'' is finite and one considers only stem extensions, then there is a largest size for such a group ''C'', and for every ''C'' of that size the subgroup ''K'' is isomorphic to the Schur multiplier of ''G''. If the finite group ''G'' is moreover perfect, then ''C'' is unique up to isomorphism and is itself perfect. Such ''C'' are often called universal perfect central extensions of ''G'', or covering group (as it is a discrete analog of the
universal 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 ...
in topology). If the finite group ''G'' is not perfect, then its Schur covering groups (all such ''C'' of maximal order) are only
isoclinic In mathematics, specifically group theory, isoclinism is an equivalence relation on groups which generalizes isomorphism. Isoclinism was introduced by to help classify and understand p-groups, although it is applicable to all groups. Isoclinism al ...
. It is also called more briefly a universal central extension, but note that there is no largest central extension, as 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 ta ...
of ''G'' and an
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 commut ...
form a central extension of ''G'' of arbitrary size. Stem extensions have the nice property that any lift of a generating set of ''G'' is a generating set of ''C''. If the group ''G'' is presented in terms of a
free group In mathematics, the free group ''F'S'' over a given set ''S'' consists of all words that can be built from members of ''S'', considering two words to be different unless their equality follows from the group axioms (e.g. ''st'' = ''suu''−1' ...
''F'' on a set of generators, and 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 i ...
''R'' generated by a set of relations on the generators, so that G \cong F/R, then the covering group itself can be presented in terms of ''F'' but with a smaller normal subgroup ''S'', that is, C\cong F/S. Since the relations of ''G'' specify elements of ''K'' when considered as part of ''C'', one must have S \le ,R/math>. In fact if ''G'' is perfect, this is all that is needed: ''C'' ≅ 'F'',''F'' 'F'',''R''and M(''G'') ≅ ''K'' ≅ ''R''/ 'F'',''R'' Because of this simplicity, expositions such as handle the perfect case first. The general case for the Schur multiplier is similar but ensures the extension is a stem extension by restricting to the derived subgroup of ''F'': M(''G'') ≅ (''R'' ∩ 'F'', ''F''/ 'F'', ''R'' These are all slightly later results of Schur, who also gave a number of useful criteria for calculating them more explicitly.


Relation to efficient presentations

In
combinatorial group theory In mathematics, combinatorial group theory is the theory of free groups, and the concept of a presentation of a group by generators and relations. It is much used in geometric topology, the fundamental group of a simplicial complex having in a nat ...
, a group often originates from a
presentation A presentation conveys information from a speaker to an audience. Presentations are typically demonstrations, introduction, lecture, or speech meant to inform, persuade, inspire, motivate, build goodwill, or present a new idea/product. Presenta ...
. One important theme in this area of mathematics is to study presentations with as few relations as possible, such as one relator groups like
Baumslag–Solitar group In the mathematical field of group theory, the Baumslag–Solitar groups are examples of two-generator one-relator groups that play an important role in combinatorial group theory and geometric group theory as (counter)examples and test-cases. ...
s. These groups are infinite groups with two generators and one relation, and an old result of Schreier shows that in any presentation with more generators than relations, the resulting group is infinite. The borderline case is thus quite interesting: finite groups with the same number of generators as relations are said to have a
deficiency A deficiency is generally a lack of something. It may also refer to: *A deficient number, in mathematics, a number ''n'' for which ''σ''(''n'') < 2''n'' * efficient group In mathematics, a presentation is one method of specifying a group. A presentation of a group ''G'' comprises a set ''S'' of generators—so that every element of the group can be written as a product of powers of some of these generators—and ...
'' is one where the Schur multiplier requires this number of generators. A fairly recent topic of research is to find efficient presentations for all finite simple groups with trivial Schur multipliers. Such presentations are in some sense nice because they are usually short, but they are difficult to find and to work with because they are ill-suited to standard methods such as
coset enumeration In mathematics, coset enumeration is the problem of counting the cosets of a subgroup ''H'' of a group ''G'' given in terms of a presentation. As a by-product, one obtains a permutation representation for ''G'' on the cosets of ''H''. If ''H'' has ...
.


Relation to topology

In
topology In mathematics, topology (from the Greek language, Greek words , and ) is concerned with the properties of a mathematical object, geometric object that are preserved under Continuous function, continuous Deformation theory, deformations, such ...
, groups can often be described as finitely presented groups and a fundamental question is to calculate their integral homology H_n(G, \Z). In particular, the second homology plays a special role and this led
Heinz Hopf Heinz Hopf (19 November 1894 – 3 June 1971) was a German mathematician who worked on the fields of topology and geometry. Early life and education Hopf was born in Gräbschen, Germany (now , part of Wrocław, Poland), the son of Elizabeth ( ...
to find an effective method for calculating it. The method in is also known as Hopf's integral homology formula and is identical to Schur's formula for the Schur multiplier of a finite group: : H_2(G, \Z) \cong (R \cap , F/ , R/math> where G \cong F/R and ''F'' is a free group. The same formula also holds when ''G'' is a perfect group. The recognition that these formulas were the same led
Samuel Eilenberg Samuel Eilenberg (September 30, 1913 – January 30, 1998) was a Polish-American mathematician who co-founded category theory (with Saunders Mac Lane) and homological algebra. Early life and education He was born in Warsaw, Kingdom of Poland to a ...
and
Saunders Mac Lane Saunders Mac Lane (4 August 1909 – 14 April 2005) was an American mathematician who co-founded category theory with Samuel Eilenberg. Early life and education Mac Lane was born in Norwich, Connecticut, near where his family lived in Taftvill ...
to the creation of cohomology of groups. In general, :H_2(G, \Z) \cong \bigl( H^2(G, \Complex^) \bigr)^* where the star denotes the algebraic dual group. Moreover, when ''G'' is finite, there is an unnatural isomorphism :\bigl( H^2(G, \Complex^) \bigr)^* \cong H^2(G, \Complex^). The Hopf formula for H_2(G) has been generalised to higher dimensions. For one approach and references see the paper by Everaert, Gran and Van der Linden listed below. A
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 universa ...
is one whose first integral homology vanishes. A
superperfect group In mathematics, in the realm of group theory, a group (mathematics), group is said to be superperfect when its first two group homology, homology groups are trivial group, trivial: ''H''1(''G'', Z) = ''H''2(''G'', Z) = 0. This is stronger than a pe ...
is one whose first two integral homology groups vanish. The Schur covers of finite perfect groups are superperfect. An
acyclic group In mathematics, an acyclic space is a nonempty topological space ''X'' in which cycles are always boundaries, in the sense of homology theory. This implies that integral homology groups in all dimensions of ''X'' are isomorphic to the correspond ...
is a group all of whose reduced integral homology vanishes.


Applications

The second algebraic K-group K2(''R'') of a commutative ring ''R'' can be identified with the second homology group ''H''2(''E''(''R''), Z) of the group ''E''(''R'') of (infinite) elementary matrices with entries in ''R''.


See also

*
Quasisimple group In mathematics, a quasisimple group (also known as a covering group) is a group that is a perfect central extension ''E'' of a simple group ''S''. In other words, there is a short exact sequence :1 \to Z(E) \to E \to S \to 1 such that E = , E/ ...
The references from Clair Miller give another view of the Schur Multiplier as the kernel of a morphism κ: G ∧ G → G induced by the commutator map.


Notes


References

* * * *
Errata
* * * * * * * * * * * * {{refend Group theory Homological algebra