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| '''Brauer's main theorems''' are three theorems in [[representation theory of finite groups]] linking the [[modular representation theory|blocks]] of a [[finite group]] (in characteristic ''p'') with those of its [[p-local subgroup|''p''-local subgroups]], that is to say, the [[normalizer]]s of its non-trivial ''p''-subgroups.
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| The second and third main theorems allow refinements of orthogonality relations for [[character theory|ordinary character]]s which may be applied in finite [[group theory]]. These do not presently admit a proof purely in terms of ordinary characters.
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| All three main theorems are stated in terms of the '''Brauer correspondence'''.
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| ==Brauer correspondence==
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| There are many ways to extend the definition which follows, but this is close to the early treatments
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| by Brauer. Let ''G'' be a finite group, ''p'' be a prime, ''F'' be a ''field'' of characteristic ''p''.
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| Let ''H'' be a subgroup of ''G'' which contains
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| :<math>QC_G(Q)</math>
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| for some ''p''-subgroup ''Q''
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| of ''G,'' and is contained in the normalizer
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| :<math>N_G(Q)</math>. | |
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| The '''Brauer homomorphism''' (with respect to ''H'') is a linear map from the center of the group algebra of ''G'' over ''F'' to the corresponding algebra for ''H''. Specifically, it is the restriction to
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| <math>Z(FG)</math> of the (linear) projection from <math>FG</math> to <math>FC_G(Q)</math> whose
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| kernel is spanned by the elements of ''G'' outside <math>C_G(Q)</math>. The image of this map is contained in
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| <math>Z(FH)</math>, and it transpires that the map is also a ring homomorphism.
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| Since it is a [[ring homomorphism]], for any block ''B'' of ''FG'', the Brauer homomorphism
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| sends the identity element of ''B'' either to ''0'' or to an idempotent element. In the latter case,
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| the idempotent may be decomposed as a sum of (mutually orthogonal) [[primitive idempotent]]s of ''Z(FH).''
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| Each of these primitive idempotents is the multiplicative identity of some block of ''FH.'' The block ''b'' of ''FH'' is said to be a '''Brauer correspondent''' of ''B'' if its identity element occurs
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| in this decomposition of the image of the identity of ''B'' under the Brauer homomorphism.
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| ==Brauer's first main theorem==
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| Brauer's first main theorem {{harvs|last=Brauer|year1=1944|year2=1956|year3=1970}} states that if <math>G</math> is a finite group a <math>D</math> is a <math>p</math>-subgroup of <math>G</math>, then there is a [[bijection]] between the set of
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| (characteristic ''p'') blocks of <math>G</math> with defect group <math>D</math> and blocks of the normalizer <math>N_G(D)</math> with
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| defect group ''D''. This bijection arises because when <math>H = N_G(D)</math>, each block of ''G''
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| with defect group ''D'' has a unique Brauer correspondent block of ''H'', which also has defect
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| group ''D''.
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| ==Brauer's second main theorem==
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| Brauer's second main theorem {{harvs|last=Brauer|year1=1944|year2=1959}} gives, for an element ''t'' whose order is a power of a prime ''p'', a criterion for a (characteristic ''p'') block of <math>C_G(t)</math> to correspond to a given block of <math>G</math>, via ''generalized decomposition numbers''. These are the coefficients which occur when the restrictions of ordinary characters of <math>G</math> (from the given block) to elements of the form ''tu'', where ''u'' ranges over elements of order prime to ''p'' in <math>C_G(t)</math>, are written as linear combinations of the irreducible [[modular representation theory|Brauer character]]s of <math>C_G(t)</math>. The content of the theorem is that it is only necessary to use Brauer characters from blocks of <math>C_G(t)</math> which are Brauer correspondents of the chosen block of ''G''.
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| ==Brauer's third main theorem==
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| Brauer's third main theorem {{harv|Brauer|1964|loc=theorem3}} states that when ''Q'' is a ''p''-subgroup of the finite group ''G'',
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| and ''H'' is a subgroup of ''G,'' containing <math>QC_G(Q)</math>, and contained in <math>N_G(Q)</math>, | |
| then the [[modular representation theory|principal block]] of ''H'' is the only Brauer correspondent of the principal block of ''G'' (where the blocks referred to are calculated in characteristic ''p'').
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| ==References==
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| *{{Citation | last1=Brauer | first1=R. | author1-link=Richard Brauer | title=On the arithmetic in a group ring | jstor=87919 | mr=0010547 | year=1944 | journal=[[Proceedings of the National Academy of Sciences|Proceedings of the National Academy of Sciences of the United States of America]] | issn=0027-8424 | volume=30 | pages=109–114}}
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| *{{Citation | last1=Brauer | first1=R. | author1-link=Richard Brauer | title=On blocks of characters of groups of finite order I | jstor=87578 | mr=0016418 | year=1946 | journal=[[Proceedings of the National Academy of Sciences|Proceedings of the National Academy of Sciences of the United States of America]] | issn=0027-8424 | volume=32 | pages=182–186}}
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| *{{Citation | last1=Brauer | first1=R. | author1-link=Richard Brauer | title=On blocks of characters of groups of finite order. II | jstor=87838 | mr=0017280 | year=1946 | journal=[[Proceedings of the National Academy of Sciences|Proceedings of the National Academy of Sciences of the United States of America]] | issn=0027-8424 | volume=32 | pages=215–219}}
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| *{{Citation | last1=Brauer | first1=R. | author1-link=Richard Brauer | title=Zur Darstellungstheorie der Gruppen endlicher Ordnung | doi=10.1007/BF01187950 | mr=0075953 | year=1956 | journal=[[Mathematische Zeitschrift]] | issn=0025-5874 | volume=63 | pages=406–444}}
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| *{{Citation | last1=Brauer | first1=R. | author1-link=Richard Brauer | title=Zur Darstellungstheorie der Gruppen endlicher Ordnung. II | doi=10.1007/BF01162934 | mr=0108542 | year=1959 | journal=[[Mathematische Zeitschrift]] | issn=0025-5874 | volume=72 | pages=25–46}}
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| *{{Citation | last1=Brauer | first1=R. | author1-link=Richard Brauer | title=Some applications of the theory of blocks of characters of finite groups. I | doi=10.1016/0021-8693(64)90031-6 | mr=0168662 | year=1964 | journal=[[Journal of Algebra]] | issn=0021-8693 | volume=1 | pages=152–167}}
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| *{{Citation | last1=Brauer | first1=R. | author1-link=Richard Brauer | title=On the first main theorem on blocks of characters of finite groups. | url=http://projecteuclid.org/euclid.ijm/1256053174 | mr=0267010 | year=1970 | journal=Illinois Journal of Mathematics | issn=0019-2082 | volume=14 | pages=183–187}}
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| *{{Citation | last1=Dade | first1=Everett C. | author1-link=Everett C. Dade | editor1-last=Powell | editor1-first=M. B. | editor2-last=Higman | editor2-first=Graham | editor2-link=Graham Higman | title=Finite simple groups. Proceedings of an Instructional Conference organized by the London Mathematical Society (a NATO Advanced Study Institute), Oxford, September 1969. | publisher=[[Academic Press]] | location=Boston, MA | isbn=978-0-12-563850-0 | mr=0360785 | year=1971 | chapter=Character theory pertaining to finite simple groups | pages=249–327}} gives a detailed proof of the Brauer's main theorems.
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| *{{eom|id=b/b120440|first=H.|last= Ellers|title=Brauer's first main theorem}}
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| *{{eom|id=b/b120450|first=H.|last= Ellers|title=Brauer height-zero conjecture}}
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| *{{eom|id=b/b120460|first=H.|last= Ellers|title=Brauer's second main theorem}}
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| *{{eom|id=b/b120470|first=H.|last= Ellers|title=Brauer's third main theorem}}
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| * [[Walter Feit]], ''The representation theory of finite groups.'' North-Holland Mathematical Library, 25. North-Holland Publishing Co., Amsterdam-New York, 1982. xiv+502 pp. ISBN 0-444-86155-6
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| [[Category:Representation theory of finite groups]]
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| [[Category:Theorems in representation theory]]
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