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| In mathematical [[group theory]], the '''Schur multiplier''' or '''Schur multiplicator''' is the second [[group homology|homology group]] ''H''<sub>2</sub>(''G'', '''Z''') of a group ''G''. It was introduced by {{harvs|txt|first=Issai|last=Schur|authorlink=Issai Schur|year=1904}} in his work on [[projective representation]]s.
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| ==Examples and properties==
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| The Schur multiplier M(''G'') of a finite group ''G'' is a finite [[abelian group]] whose [[Periodic group|exponent]] divides the order of ''G''. If a [[Sylow subgroup|Sylow ''p''-subgroup]] of ''G'' is cyclic for some ''p'', then the order of M(''G'') is not divisible by ''p''. In particular, if all [[Sylow subgroup|Sylow ''p''-subgroups]] of ''G'' are cyclic, then M(''G'') is trivial.
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| For instance, the Schur multiplier of the [[nonabelian group of order 6]] is the [[trivial group]] since every Sylow subgroup is cyclic. The Schur multiplier of 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]] is trivial, but the Schur multiplier of [[dihedral group|dihedral 2-groups]] has order 2.
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| The Schur multipliers of the finite [[simple group]]s are given at the [[list of finite simple groups]]. The [[covering groups of the alternating and symmetric groups]] are of considerable recent interest.
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| {{anchor|Schur cover}}
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| ==Relation to projective representations==
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| [[File:Projective-representation-lifting.svg|225px|thumb|A [[projective representation]] of ''G'' can be pulled back to a [[linear representation]] of a [[central extension (mathematics)|central extension]] ''C'' of ''G.'']]
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| Schur's original motivation for studying the multiplier was to classify [[projective representation]]s of a group, and the modern formulation of his definition is the second [[group cohomology|cohomology group]] ''H''<sup>2</sup>(''G'', '''C'''<sup>×</sup>). A projective representation is much like a [[group representation]] except that instead of a homomorphism into the [[general linear group]] GL(''n'', '''C'''), one takes a homomorphism into the [[projective general linear group]] PGL(''n'', '''C'''). In other words, a projective representation is a representation modulo the [[center of a group|center]].
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| {{harvs|last=Schur|author1-link=Issai Schur|txt|year=1904|year2=1907}} 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 [[list of finite simple groups|finite simple groups]] are known, and each is an example of a [[quasisimple group]]. The Schur cover of a [[perfect group]] is uniquely determined up to isomorphism, but the Schur cover of a general finite group is only determined up to [[isoclinism]].
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| ==Relation to central extensions==
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| The study of such covering groups led naturally to the study of [[central extension (mathematics)|central]] and '''stem extensions'''.
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| A [[central extension (mathematics)|central extension]] of a group ''G'' is an extension
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| :1 → ''K'' → ''C'' → ''G'' → 1
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| where ''K'' ≤ Z(''C'') is a [[subgroup]] of the [[center (group theory)|center]] of ''C''.
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| A '''stem extension''' of a group ''G'' is an extension
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| :1 → ''K'' → ''C'' → ''G'' → 1 | |
| where ''K'' ≤ Z(''C'') ∩ ''C''′ is a subgroup of the intersection of the center of ''C'' and the [[derived subgroup]] of ''C''; this is more restrictive than central.<ref>Rotman (1994) p.553</ref>
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| 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 group|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]] in topology). If the finite group ''G'' is not perfect, then its Schur covering groups (all such ''C'' of maximal order) are only [[isoclinic]].
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| It is also called more briefly a '''universal central extension''', but note that there is no largest central extension, as the [[direct product of groups|direct product]] of ''G'' and an [[abelian group]] form a central extension of ''G'' of arbitrary size.
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| 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 [[presentation of a group|presented]] in terms of a [[free group]] ''F'' on a set of generators, and a [[normal subgroup]] ''R'' generated by a set of relations on the generators, so that ''G'' ≅ ''F''/''R'', then the covering group itself can be presented in terms of ''F'' but with a smaller normal subgroup ''S'', ''C'' ≅ ''F''/''S''. Since the relations of ''G'' specify elements of ''K'' when considered as part of ''C'', one must have ''S'' ≤ [''F'',''R''].
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| 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 {{harv|Aschbacher|2000|loc=§33}} 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.
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| ==Relation to efficient presentations== | |
| In [[combinatorial group theory]], a group often originates from a [[presentation of a group|presentation]]. 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]]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 (group theory)|deficiency]] zero. For a group to have deficiency zero, the group must have a trivial Schur multiplier because the minimum number of generators of the Schur multiplier is always less than or equal to the difference between the number of relations and the number of generators, which is the negative deficiency. An ''[[efficient group]]'' is one where the Schur multiplier requires this number of generators.<ref>Johnson & Robertson (1979) pp.275–289</ref>
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| 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 [[Todd–Coxeter algorithm|coset enumeration]].
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| ==Relation to topology==
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| In [[topology]], groups can often be described as finitely [[presentation of a group|presented]] groups and a fundamental question is to calculate their integral homology ''H<sub>n</sub>''(''G'', '''Z'''). In particular, the second homology plays a special role and this led [[Heinz Hopf|Hopf]] to find an effective method for calculating it. The method in {{harv|Hopf|1942}} is also known as '''Hopf's integral homology formula''' and is identical to Schur's formula for the Schur multiplier of a finite, finitely presented group: | |
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| :<math> H_2(G, \mathbf{Z}) \cong (R \cap [F, F])/[F, R]</math>
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| where ''G'' ≅ ''F''/''R'' and ''F'' is a free group. The same formula also holds when ''G'' is a perfect group.<ref>Rosenberg (1994) Theorems 4.1.3, 4.1.19</ref>
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| The recognition that these formulas were the same led [[Samuel Eilenberg|Eilenberg]] and [[Saunders Mac Lane|Mac Lane]] to the creation of [[cohomology of groups]]. In general,
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| :<math>H_2(G, \mathbf{Z}) \cong \bigl( H^2(G, \mathbf{C}^\times) \bigr)^* </math> | |
| where the star denotes the algebraic dual group. Moreover when ''G'' is finite, there is an [[Natural transformation|unnatural]] isomorphism
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| :<math>\bigl( H^2(G, \mathbf{C}^\times) \bigr)^* \cong H^2(G, \mathbf{C}^\times).</math>
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| The Hopf formula for ''H''<sub>2</sub>(''G'') has been generalised to higher dimensions. For one approach and references see the paper by Everaert, Gran and Van der Linden listed below.
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| A [[perfect group]] is one whose first integral homology vanishes. A [[superperfect group]] is one whose first two integral homology groups vanish. The Schur covers of finite perfect groups are superperfect. An [[acyclic group]] is a group all of whose reduced integral homology vanishes.
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| ==Applications==
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| The [[Algebraic K-theory#K2|second algebraic K-group]] K<sub>2</sub>(''R'') of a commutative ring ''R'' can be identified with the second homology group ''H''<sub>2</sub>(''E''(''R''), '''Z''') of the group ''E''(''R'') of (infinite) [[elementary matrix|elementary matrices]] with entries in ''R''.<ref>Rosenberg (1994) Corollary 4.2.10</ref>
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| ==See also==
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| * [[Quasisimple group]]
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| 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.
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| ==References==
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| {{reflist}}
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| * {{Citation | last1=Aschbacher | first1=Michael | author1-link=Michael Aschbacher | title=Finite group theory | publisher=[[Cambridge University Press]] | edition=2nd | series=Cambridge Studies in Advanced Mathematics | isbn=978-0-521-78145-9 | mr=1777008 | zbl=0997.20001 | year=2000 | volume=10}}
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| * {{Citation | last1=Hopf | first1=Heinz | author1-link=Heinz Hopf | title=Fundamentalgruppe und zweite Bettische Gruppe | doi=10.1007/BF02565622 | mr=0006510 | zbl=0027.09503 | year=1942 | journal=Commentarii Mathematici Helvetici | issn=0010-2571 | volume=14 | pages=257–309}}
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| *{{citation | first1=D.L. | last1=Johnson | first2=E.L. | last2=Robertson | chapter=Finite groups of deficiency zero | editor1-first=C.T.C. | editor1-last=Wall | editor-link=C. T. C. Wall | title=Homological Group Theory | series=London Mathematical Society Lecture Note Series | volume=36 | year=1979 | publisher=[[Cambridge University Press]] | isbn=0-521-22729-1 | zbl=0423.20029 }}
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| *{{eom|first=L.V.|last= Kuzmin|id=S/s083460|title=Schur multiplicator}}
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| *{{Citation | last1=Rosenberg | first1=Jonathan | authorlink=Jonathan Rosenberg (mathematician)| | title=Algebraic K-theory and its applications | url=http://books.google.com/books?id=TtMkTEZbYoYC | publisher=[[Springer-Verlag]] | location=Berlin, New York | series=[[Graduate Texts in Mathematics]] | isbn=978-0-387-94248-3 | mr=1282290 | zbl=0801.19001 | postscript=[http://www-users.math.umd.edu/~jmr/KThy_errata2.pdf Errata] | year=1994 | volume=147}}
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| * {{Citation | last1=Rotman | first1=Joseph J. | title=An introduction to the theory of groups | publisher=[[Springer-Verlag]] | location=Berlin, New York | isbn=978-0-387-94285-8 | year=1994}}
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| * {{Citation | last1=Schur | first1=J. | author1-link=Issai Schur | title=Über die Darstellung der endlichen Gruppen durch gebrochene lineare Substitutionen. | url=http://resolver.sub.uni-goettingen.de/purl?GDZPPN002165511 | language=German | jfm=35.0155.01 | year=1904 | journal=[[Journal für die reine und angewandte Mathematik]] | issn=0075-4102 | volume=127 | pages=20–50}}
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| * {{Citation | last1=Schur | first1=J. | author1-link=Issai Schur | title=Untersuchungen über die Darstellung der endlichen Gruppen durch gebrochene lineare Substitutionen. | url=http://resolver.sub.uni-goettingen.de/purl?GDZPPN00216633X | language=German | jfm=38.0174.02 | year=1907 | journal=[[Journal für die reine und angewandte Mathematik]] | issn=0075-4102 | volume=132 | pages=85–137}}
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| * {{citation|title=Review: F. Rudolf Beyl and Jürgen Tappe, Group extensions, representations, and the Schur multiplicator|first=Wilberd|last=Van der Kallen|volume=10|number=2|year=1984|pages=330–333|journal=Bulletin of the American Mathematical Society|url=http://projecteuclid.org/euclid.bams/1183551591}}
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| * {{Citation | last1=Wiegold | first1=J. | author1-link=James Wiegold | title=Groups–St. Andrews 1981 (St. Andrews, 1981) | publisher=[[Cambridge University Press]] | series=London Math. Soc. Lecture Note Ser. | mr=679156 | zbl=0502.20003 | year=1982 | volume=71 | chapter=The Schur multiplier: an elementary approach | pages=137–154}}
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| * {{citation | last=Miller | first=Clair | title=The second homology of a group | journal=Proc. American Math. Soc. | volume=3 | year=1952 | pages=588–595 | zbl=0047.25703 }}
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| * {{citation | first=R.K. | last=Dennis | title=In search of new "Homology" functors having a close relationship to K-theory | publisher=Cornell University | year=1976 }}
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| * {{citation | first1=R. | last1=Brown | first2=D.L. | last2=Johnson | first3=E.F. | last3=Robertson | title=Some computations of non-abelian tensor products of groups | journal= J. Algebra | volume=111 | year=1987 | pages=177–202 | zbl=0626.20038 }}
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| *{{citation | first1=G.J. | last1=Ellis | first2=F. | last2=Leonard | title=Computing Schur multipliers and tensor products of finite groups | journal=Proc. Royal Irish Acad. |volume=95A | year=1995 | pages=137–147 | zbl=0863.20010 | issn=0035-8975 }}
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| * {{citation | first=G.J. | last=Ellis | title=The Schur multiplier of a pair of groups | journal=Appl. Categ. Structures | volume=6 | year=1998 | pages=355–371 | zbl=0948.20026 }}
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| * {{citation | last1=Eick | first1=Bettina | last2=Nickel | first2=Werner | title=Computing the Schur multiplicator and the nonabelian tensor square of a polycyclic group | journal=J. Algebra | volume=320 | year=2008 | pages=927–944 | zbl=1163.20022 }}
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| * {{citation | last1=Everaert | first1=Tomas | last2=Gran | first2=Marino | last3=Van der Linden | first3=Tim | title=Higher Hopf formulae for homology via Galois theory | journal=Adv. Math. | volume=217 | year=2008 | number=5 | pages=2231–2267 | zbl=1140.18012 }}
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| [[Category:Group theory]]
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| [[Category:Homological algebra]]
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