the green corr and brauer char

7/28/2019 The Green Corr and Brauer Char

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THE GREEN CORRESPONDENCE ND BRAUER'S- --- CHnRACTERIZATIoN oF cHARACTERS


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THE GREEN CORRESPONDENCEND BRAUER'SCHARACTERIZATION F CHARACTERSJ . A L P E R I N

An analogy between the role of vertices n the Grecn correspondence ndelementary ubgroups n Brauer 's nduct ion heorcmhas bccn madeby J. A. Greeni l ] Wc shal l show that thcrc is. in fact . a direct connect ion: he Greencorrespondencempl iesBrauer 's heorem.

Le t us fix some notation. Let K be a number field which is a splitting field or thesubgroups f the nonident i ty roup G. For eachpr imedivisorp of the order of G letR, be the completionof th e integersR of K wit h respect o a prime divisor of p in R.We shall implicitly assume hat all RoG-modules re finitely generatedand free asRn-modules.We shal l say that an RnG-module s p- local f i t is the direct sum ofmodules nduced rom p-l


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THE GREEN CORRESPONDENCE r39Proof. It sufficeso prove he result or theprincipalcharacter oof G nasmuchas the integral inearcombinations f charactersnduced rom a subgroup I of Gform an ideal n the ring of all generalizedharacters f G. For eachprime p let1o Ar+Oo as n thepreviousesult.Hence,c : fl (Ao+Oo) ndeach erm n the

expansion, ave he product of the O' is clearlyur i int.g.ul linearcombinationofcharactersnduced rom local subgroups. owever, he productof the Oo vanisheson eachnon-identity lement f G so t is a multipleof the egular haracter nd so sinduced rom any subgroupof G.Conoru,Rv3 (Brauer'snductionTheorem). Euerycharacter f G is an integralIinearcombination f charactersnducedrom elementaryubgroupsf G.Of course,as usual, this immediately mplies Brauer's characteizationofcharacters.Proof. Let G be a counterexample f minimal order. We shall analyze hestructureof G in a numberof stepsand finally reacha contradiction.(1) The principal character 1o is not an integral linear combination ofcharactersnduced rom elementary ubgroups.Indeed, he collectionof all such inearcombinationss an ideal n the ring of allgeneralizedharacters f G.(2) There s a primep such hat Oo(G)* l.If every ocal subgroupof G is proper hen Corollary 3 and the minimalityof G

would contradict he fact that G is a counterexampleo the theorem.(3) Everyproperhomomorphicmageof G is elementary.Supposehat G s a properhomomorphicmageof G andG s not elementary. ythe minimalityof G, theprincipalcharacter f G s an integral inearcombinationofcharactersnduced rom propersubgroups f G; hence hesame s true n G. But thetheoremholds n everypropersubgroupof G so againwe havea contradiction.This allows us to give a detaileddescriptionof G. Let p be a prime such hatOr(G)+ I and et N be a minimal normalsubgroup f G with N a p-group, o N iselementary belian.Hence,G/N is elementary.Thus,G/N is the directproductof aq-groupQIN for a prime q, and a cyclic4'-groupC/N.)(4) G is not nilpotent.If G is nilpotent hen,sinceG is not elementary, hasa homomorphicmageofth e form Z,xZ,xZ"xZ" for distinctprimes and s. Hence,G is isomorphicwiththisgroup,by (3).Let R andSbe the Sylow -subgroup nd Sylows-subgroup f G.The character1* is the sum of the charactersnduced rom the principalcharactersof eachof thesubgroups f order r in R minus he regular haracter. ence, 1o s anintegral inearcombinationof charactersnducedup from elementary ubgroups fthe orm S andRox S, whereRo s a subgroup f order in R. The character 1ohasa similarexpression o that we havea contradiction o (1).(5) N is the uniqueminimalnormal subgroupof G.Suppose hat M is another such subgroup so M n N : 1. Hence, Gisomorphicwith a subgroup of GlMxGlN. However,each of these actorsnilpotent,by (3),so that G is also,contradicting4).

ISI S


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140 THE CREEN CORRESPONDENCE(6) N: Op(G).certainly N c oe(G).since G/N is nilpotent wehave hat G/or(N) is a p'-group.Hence, if oe(G) is elementary abelian then Maschke's theorem yields thatOe(N) : N. If o?(G) s not elementary belian hen the Frattini subgroupD(o,(G))of oo(G) s not I so mustcontainN, by (5).But then oop)lD(oo(G)) is a centralfactorof G so that N will be also.Hence,N is a central actor and GIN s nilpotent sowe have contradicted(4).The Schur-Zassenhausheorem now applies and G : KN is a semi-directproduct of N and a subgroupK. The uniqueness f N now givesus that N is not acentral factor of G; hence,no non-trivial characterof N is stabilizedby G. Thecharacter(1*)c of G inducedby the principal characterof K is the sum of r.oand

characters f G whose estrictions o N do not involve 1". Hence,by Clifford theory,eachofthesecharacterss nduced rom a propersubgroup f G. This shows hat 1ois a linear combination of charactersnduced rom proper subgroupsand we haveafinal contradiction.

ReferencesJ. A. Green, Axiomatic representationheory for finite groups",J Pure Appl. Alg. l (1971\,41J7.J. A. Green,*A transfer heorem or modular representations",. Alg. | (1964\73-U.

Departmentof Mathematics,University of Chicago,Chicago,Illinois 60637.U.S.A.

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the green corr and brauer char

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