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Idempotent Elements in Blocks of p-Solvable Groups

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Title: Idempotent Elements in Blocks of p-Solvable Groups
Physical Description: 1 online resource (71 p.)
Language: english
Creator: Raney, Lee S
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2013

Subjects

Subjects / Keywords: algebra -- basic -- block -- brauer -- category -- character -- defect -- dimension -- finite -- group -- idempotent -- mathematics -- matrix -- modular -- module -- morita -- prime -- representation -- solvable -- support
Mathematics -- Dissertations, Academic -- UF
Genre: Mathematics thesis, Ph.D.
bibliography   ( marcgt )
theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
born-digital   ( sobekcm )
Electronic Thesis or Dissertation

Notes

Abstract: Given a finite group G and an algebraically closed field F of prime characteristic p, the p-blocks of G are the indecomposable two-sided ideals of the group algebra FG. Each block of G is determined by a primitive idempotent of the center Z(FG) of the group algebra. When studying a ring R in general, one is interested in the category of R-modules. Given a finite dimensional algebra A, its basic algebra is another algebra B which is simplest among those whose category of B-modules is equivalent to the category of A-modules. In this work, we investigate properties of idempotent elements in blocks of p-solvable groups, and we describe a method for calculating the basic algebra of such a block. We conclude with a software implementation of this method, and we present some empirical data obtained using this implementation.
General Note: In the series University of Florida Digital Collections.
General Note: Includes vita.
Bibliography: Includes bibliographical references.
Source of Description: Description based on online resource; title from PDF title page.
Source of Description: This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
Statement of Responsibility: by Lee S Raney.
Thesis: Thesis (Ph.D.)--University of Florida, 2013.
Local: Adviser: Turull, Alexandre.
Electronic Access: RESTRICTED TO UF STUDENTS, STAFF, FACULTY, AND ON-CAMPUS USE UNTIL 2015-05-31

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Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2013
System ID: UFE0045384:00001

Permanent Link: http://ufdc.ufl.edu/UFE0045384/00001

Material Information

Title: Idempotent Elements in Blocks of p-Solvable Groups
Physical Description: 1 online resource (71 p.)
Language: english
Creator: Raney, Lee S
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2013

Subjects

Subjects / Keywords: algebra -- basic -- block -- brauer -- category -- character -- defect -- dimension -- finite -- group -- idempotent -- mathematics -- matrix -- modular -- module -- morita -- prime -- representation -- solvable -- support
Mathematics -- Dissertations, Academic -- UF
Genre: Mathematics thesis, Ph.D.
bibliography   ( marcgt )
theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
born-digital   ( sobekcm )
Electronic Thesis or Dissertation

Notes

Abstract: Given a finite group G and an algebraically closed field F of prime characteristic p, the p-blocks of G are the indecomposable two-sided ideals of the group algebra FG. Each block of G is determined by a primitive idempotent of the center Z(FG) of the group algebra. When studying a ring R in general, one is interested in the category of R-modules. Given a finite dimensional algebra A, its basic algebra is another algebra B which is simplest among those whose category of B-modules is equivalent to the category of A-modules. In this work, we investigate properties of idempotent elements in blocks of p-solvable groups, and we describe a method for calculating the basic algebra of such a block. We conclude with a software implementation of this method, and we present some empirical data obtained using this implementation.
General Note: In the series University of Florida Digital Collections.
General Note: Includes vita.
Bibliography: Includes bibliographical references.
Source of Description: Description based on online resource; title from PDF title page.
Source of Description: This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
Statement of Responsibility: by Lee S Raney.
Thesis: Thesis (Ph.D.)--University of Florida, 2013.
Local: Adviser: Turull, Alexandre.
Electronic Access: RESTRICTED TO UF STUDENTS, STAFF, FACULTY, AND ON-CAMPUS USE UNTIL 2015-05-31

Record Information

Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2013
System ID: UFE0045384:00001


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IDEMPOTENTELEMENTSINBLOCKSOFp-SOLVABLEGROUPSByLEESTEPHENRANEYADISSERTATIONPRESENTEDTOTHEGRADUATESCHOOLOFTHEUNIVERSITYOFFLORIDAINPARTIALFULFILLMENTOFTHEREQUIREMENTSFORTHEDEGREEOFDOCTOROFPHILOSOPHYUNIVERSITYOFFLORIDA2013

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c2013LeeStephenRaney 2

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Tomygrandmother 3

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ACKNOWLEDGMENTS First,Iwouldliketothankmyadvisor,AlexandreTurull,withoutwhosewisdomandinnitepatiencethisworkwouldnothavebeenpossible.IwouldalsoliketothankPeterSinforsharingwithmemanyvaluableconversationsonthistopic,andIthankJorgeMartinez,KevinKeating,andLyleBrennerforservingonmycommittee,askingmanyinsightfulquestions,andprovidingworthwhilesuggestions.Mydeepestthanksgotomyfamilyforalwayssupportingme,andtoJill,theloveofmylife,forgoingthroughallofthisbymyside.IalsowishtothankOrchidandJasper.Finally,Ithankcoffee. 4

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TABLEOFCONTENTS page ACKNOWLEDGMENTS .................................. 4 LISTOFTABLES ...................................... 6 ABSTRACT ......................................... 7 CHAPTER 1INTRODUCTION ................................... 8 2PRELIMINARYREPRESENTATIONTHEORY .................. 12 2.1Algebras,Modules,andIdempotents ..................... 12 2.2CharactersandBlocksofFiniteGroups ................... 20 2.3CharactersandBlocksofp-SolvableGroups ................ 25 3BLOCKIDEMPOTENTSFORp-SOLVABLEGROUPS .............. 29 3.1BlocksofFullDefect .............................. 29 3.2BlocksContainingaLinearBrauerCharacter ................ 34 4BASICALGEBRASOFBLOCKSOFp-SOLVABLEGROUPS .......... 40 4.1TheBasicAlgebraofaBlockofap-SolvableGroup ............ 40 4.2Blocksofp-NilpotentGroups ......................... 41 4.3BasicIdempotentsforthePrincipalBlockWithTwoIrreducibleBrauerCharactersofap-SolvableGroup ...................... 48 5IMPLEMENTATIONINGAP ............................. 54 5.1TheAlgorithm .................................. 54 5.2SourceCode .................................. 56 5.3SampleOutput ................................. 62 REFERENCES ....................................... 69 BIOGRAPHICALSKETCH ................................ 71 5

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LISTOFTABLES Table page 5-1DimensionsofBlocksandTheirBasicAlgebrasinSmallSolvableGroups ... 63 6

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AbstractofDissertationPresentedtotheGraduateSchooloftheUniversityofFloridainPartialFulllmentoftheRequirementsfortheDegreeofDoctorofPhilosophyIDEMPOTENTELEMENTSINBLOCKSOFp-SOLVABLEGROUPSByLeeStephenRaneyMay2013Chair:AlexandreTurullMajor:MathematicsGivenanitegroupGandanalgebraicallyclosedeldFofprimecharacteristicp,thep-blocksofGaretheindecomposabletwo-sidedidealsofthegroupalgebraFG.EachblockofGisdeterminedbyaprimitiveidempotentofthecenterZ(FG)ofthegroupalgebra.WhenstudyingaringRingeneral,oneisinterestedinthecategoryofR-modules.GivenanitedimensionalalgebraA,itsbasicalgebraisanotheralgebraBwhichissimplestamongthosewhosecategoryofB-modulesisequivalenttothecategoryofA-modules.Inthiswork,weinvestigatepropertiesofidempotentelementsinblocksofp-solvablegroups,andwedescribeamethodforcalculatingthebasicalgebraofsuchablock.Weconcludewithasoftwareimplementationofthismethod,andwepresentsomeempiricaldataobtainedusingthisimplementation. 7

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CHAPTER1INTRODUCTIONThestudyofdiscretemathematicalstructuresplaysacrucialroleinmodernmathematics,notonlybecauseofitsintrinsicinterest,butalsobecauseofitsapplicationstoscience.Attheheartofthestudyofsuchstructuresisnitegrouptheory.Groupsareamathematicalrealizationofthenotionofsymmetry,andtheirstudyisafundamentalpartofabstractalgebraandmathematicsingeneral.Onewaytostudynitegroupsistoexaminethemannerinwhichtheycanactuponothervariousmathematicalobjects.Grouprepresentationtheoryisthestudyofactionsofgroupsonvectorspaces.Modularrepresentationtheoryisthestudyofrepresentationsofnitegroupsonnitedimensionalvectorspacesovereldsofprimecharacteristic.Blocktheoryisatoolforstudyingmodularrepresentations.Inaddition,blocktheoryshedslightontherelationshipsbetweenthemodularrepresentationsofagivennitegroupandthoseincharacteristiczeroandoftenyieldsnewinformationaboutboth.Onewaytounderstandtherepresentationtheoryofanygivennitedimensionalassociativealgebra(or,forexample,agroupalgebraofanitegroup)istounderstanditscategoryofmodules.Suchcategorieswerestudiedin[ 16 ].Inthispaper,MoritaprovedthattheisomorphismclassofthebasicalgebraofanalgebraAdeterminestheequivalenceclassofthecategorywhoseobjectsareA-modulesandwhosemorphismsareA-modulehomomorphisms.Thisfacthasinspiredagreatdealofcurrentinterestinthecalculationofbasicalgebras.Inthisdissertation,weprovideaconcretetreatmentofblockidempotentsofp-solvablegroupsandatreatmentofthebasicalgebraofablockofap-solvablegroupatthelevelofidempotentsandgroupcharacters.Amainresultisamethodforcomputingthebasicalgebraofablockofap-solvablegroupbysystematicallycombiningthetheoryofMoritain[ 16 ]andFong'stheoryofcharactersandblocksofp-solvablegroups,originallydevelopedin[ 7 ].Weconcludebydiscussinganalgorithm 8

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forcomputingthebasicalgebraofablockofp-solvablegroupanditsimplementationintheGAP[ 8 ]computersoftware.InChapter2,wediscussthenecessarypreliminariesfromtherepresentationtheoryofnitegroups.Section2.1isarecollectionofstandardfactsaboutnitedimensionalalgebras,nitedimensionalmodulesoveralgebras,andidempotentelementsinalgebras.Wedenethebasicalgebraofanitedimensionalalgebraasin[ 1 ],andweshowthatthebasicalgebraofanalgebraAcanbeconstructedusingsomeidempotentelementinA.GivenalgebrasAandB,wesaythatAandBareMoritaequivalentiftheyhaveisomorphicbasicalgebras(andhence,AandBhaveequivalentmodulecategories).Hence,Moritaequivalenceisaslightlyweakernotionthanthatofisomorphism,butmanyinterestingpropertiesofalgebras(e.g.simplicity,semisimplicity,etc.)arepreservedbyMoritaequivalence.Wealsoprovesomestandardtheoremsaboutthebasicalgebraatthelevelofidempotents.Insection2.2,werecallsomefactsandnotationinvolvingcharactersandblocksofnitegroups.Ourmainreferenceforordinarycharactertheoryandcomplexrepresentationtheoryis[ 12 ],andourmainreferenceforBrauercharactertheory,andmodularrepresentationtheory,andblocktheoryis[ 17 ].Inparticular,werecalltheoremsduetoClifford,Green,andFong-Reynoldsinvolvingtherelationshipsbetweencharactersandblocksofagroupandthoseofanormalsubgroup.In[ 7 ],PaulFongrststudiedthecharactersandblocksofp-solvablegroups.Section2.3recallsthemainresultsofthetheoryofcharactersandblocksofp-solvablegroups,using[ 17 ,Chapter10]asourmainreference.Chapter3isastudyofblockidempotentsinp-solvablegroupsforblocksoffulldefect.In[ 19 ],TiedtprovedthatifBisablockoffulldefect(i.e.thedefectgroupsofBarepreciselytheSylowp-subgroupsofG),thatthesupportoftheblockidempotenteBiscontainedinOp0(G),thelargestnormalsubgroupofGwhoseorderisnotdivisiblebyp.InSection3.1,weusethisresulttoprovideacalculationoftheblockidempotenteBofablockBoffulldefectintermsoftheBrauercharacterswhichbelongtoBandalsoin 9

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termsofblockidempotentsofblocksofOp0(G)whicharecoveredbyB.InSection3.2,wespecializetoblocksBofp-solvablegroupswhichcontainalinearBrauercharacter.WeprovethatforsuchablockB,thereexistsauniquelinearG-invariantcharacterofOp0(G)thatcontrolsthestructureofB.Inparticular,theblockidempotentofBisequaltotheblockidempotentcorrespondingtothischaracter.Wealsogive(intermsoftheafforementionedcharacter)anexplicitisomorphismbetweenBandtheprincipalblock(theblockwhichcontainsthetrivialcharacter).WeconcludeSection3.2withatheoremwhichcountsthenumberofirreducibleBrauercharactersinablockwhoseonlyirreducibleBrauercharactersarelinear.Chapter4containsourtheoreticalstudyofthebasicalgebrablockBofap-solvablegroup.Section4.1containsatheoremthatyieldsamethodforconstructinganidempotentwhichdeterminesthebasicalgebraofBuptoisomorphism.AslightdrawbackofthisgeneralresultisthatifBcontainsanirreducibleBrauercharacterwhosedegreehasanontrivialp0-part,thenonemustbeabletondprimitive(notnecessarilycentral)idempotentsinordertocalculatethebasicalgebraexplicitly.However,intheeventthateveryBrauercharacterinBhasdegreeapowerofp,thebasicalgebracanbeconstructedexplicitlywithoutsearchingforprimitiveidempotents.Section4.2isdedicatedtoprovingaspecialcaseofaresultmentionedin[ 2 ]whichsaysthatifBisanilpotentblockofanarbitrarynitegroupGandBhasabeliandefectgroups,thenthebasicalgebraofBisisomorphictothegroupalgebraoveradefectgroup.Wepresentanewproofofthisresult(usingonlyelementarytechniquesofcharactertheoryandlinearalgebra)inthespecialcasethatGisap-nilpotent(andhencealsop-solvable)group.Finally,inSection4.3,weextendaresultofNinomayaandWadain[ 18 ]toreneourcalculationofthebasicalgebrainSection4.1inthecasethatBcontainsexactlytwoBrauercharacters,andatleastoneofthesecharactersislinear. 10

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Chapter5concernstheimplementationinGAPofthemethodofcomputingthebasicalgebraofablockofap-solvablegroupdescribedinSection4.1.GAP(whichstandsforGroups,Algorithms,andProgramming)isafreeandopen-sourcecomputationaldiscretealgebrasystemoriginallydevelopedatLehrstuhlDfurMathematikin1986.Sincethen,GAPhasgonethroughmanydifferentversions,anditsdevelopmentiscurrentlyjointly-coordinatedbymanydifferentauthors.Section5.1isadescriptionofthealgorithm,Section5.2istheGAPsourcecode,andSection5.3containssomeinformationaboutexamplescomputedusingthiscode.Atthe2013JointMathematicsMeetingsinSanDiego,California,KlausLuxpresentedinhistalk[ 14 ]aGAPpackagewhichisusedtostudybasicalgebrasofblocksofsimplegroups.Lux'salgorithmusesresultsofadifferentnaturesuchasthosewhichappearin[ 15 ]and[ 3 ]. 11

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CHAPTER2PRELIMINARYREPRESENTATIONTHEORY 2.1Algebras,Modules,andIdempotentsInthissection,werecallsomefundamentaldenitionsandresultsaboutalgebras,modules,andidempotents.Throughout,supposethatFisanalgebraicallyclosedeldandthatAisanitedimensional,associativeF-algebrawith1.SinceAisnitedimensional,wecandecomposetheregularleftA-moduleAasadirectsumAA=nMi=1Ai,wheretheAiareindecomposableA-submodules,calledprincipalindecomposablemodulesofA.NotethatAiisanindecomposableleftidealofA.BytheKrull-Schmidttheorem,theAiareuniqueuptoisomorphismandorderofoccurrence.Thus,theprincipalindecomposablemodules(sometimescalledprojectiveindecomposablemodulesorPIMs)ofAaredeneduptoisomorphism.Anelemente2Aiscalledanidempotentife2=e.Twoidempotents,e,f2Aareorthogonalifef=0=fe.Anidempotente2Aisprimitiveife6=0andecannotbewrittenasthesumoftwononzeroorthogonalidempotents.Thereisaninherentconnectionbetweenprimitiveidempotentsandprincipalindecomposablemodules.Wewillshow,inLemmas 2.1 and 2.2 below,thatndingasetofprimitiveorthogonalidempotentswhosesumis1inAisequivalenttondingadecompositionofAintoadirectsumofprincipalindecomposablemodules. Lemma2.1. SupposeA=nMi=1AiisadecompositionofAintoadirectsumofleftidealsofA.Foreach1in,leteibetheuniqueelementofAisuchthat1=e1++en.ThentheeiareorthogonalidempotentsofAandAi=Aeiforalli.Furthermore,ifAiisindecomposable,theneiisaprimitiveidempotent. 12

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Proof. Since1=e1++en,wehavethatforeachj,ej=nXi=1ejei2Aj.SinceA=nMi=1Aiisadirectsum,thisimpliesthatejei=ijejforalli,j.Hence,theeiaremutuallyorthogonalidempotents.Now,ifa2Ai,thena=a1=anXj=1ej2Ai,andsoAiej=0ifi6=j,sincethesumisdirect.Now,Ai=Ai1=Ai0@nXj=1ej1A=AieisinceAiej=0ifi6=j.Now,supposethatAiisindecomposableandthatei=fi+gifornonzeroorthogonalidempotentsfiandgi.ThenAi=Aei=A(fi+gi)Afi+Agi.Now,sincefiandgiareorthogonal,ifx,y2A,thenxfi+ygi=xfi(fi+gi)+ygi(fi+gi)=(xfi+ygi)(fi+gi)2A(fi+gi).Hence,Ai=Afi+Agi.Furthermore,ifx2Afi\Agi,thenx=afiforsomea2A.Butsincex2Agi,wehavethatx=xgi=afigi=0.Hence,Ai=AfiAgiisadirectsumoftwononzeroleftidealsofA,whichisacontradiction. Lemma2.2. Supposethate1,...,en2Aareorthogonalidempotentssuchthat1=e1++en.ThenA=nMi=1Aei.Furthermore,ifeiisaprimitiveidempotentforsomei,thenAeiisanindecomposableleftideal. 13

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Proof. First,wehavethatA=A1=A(e1++en)=Ae1+Aen.Ifa2Aei,thena=aei.Ifa2nXj=1j6=iAej,thena=a(e1++bei++en)=a(1)]TJ /F2 11.955 Tf 11.96 0 Td[(ei).Thus,ifa2Aei\nXj=1j6=iAej,thena=aei=a(1)]TJ /F2 11.955 Tf 11.96 0 Td[(ei)ei=a0=0.Therefore,A=nMi=1Aei.Now,supposethateiisaprimitiveidempotent,butthatAei=BiCifornonzeroleftidealsBiandCiofAei.Thenthereexistuniqueelementsfi2Biandgi2Cisuchthatei=fi+gi.Sincefi2Aei,wehavethatfiei=fi.Hence,fi=f2i=fi(ei)]TJ /F2 11.955 Tf 11.95 0 Td[(gi)=fi)]TJ /F2 11.955 Tf 11.95 0 Td[(figi,sothatfigi=0.Similarly,gifi=0.Thus,fi+gi=(fi+gi)2=f2i+g2i,sothatgiandfiaremutuallyorthogonalidempotentswhosesumisei.Sinceeiisprimitive,thisimpliesthatoneoffiandgiis0.Withoutloss,wemayassumefi=0.Thengi=ei,andCiisaleftidealofAeiwhichcontainsgi=ei.ThisimpliesthatCi=AeiwhichyieldsthatBi=0,acontradiction. CombiningLemmas 2.1 and 2.2 ,weseethatifIisaleftidealofA,thenIisaprincipalindecomposablemoduleofAifandonlyifI=Aeforsomeprimitiveidempotente2A.WewillbeparticularlyinterestedintheA-endomorphismalgebrasofsuchmodules. 14

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Lemma2.3. LetAbeanalgebraoveraeldande2Aanidempotent.ThenAeisaleftA-moduleandEndA(Ae)'(eAe)opasF-algebras. Proof. LetE=EndA(Ae).First,notethateveryf2Eiscompletelydeterminedbyf(e)sincef(ae)=af(e)foralla2A.Now,deneamap:E!(eAe)opby(f)=ef(e)eforallf2E.Then,observethatifb2Ae,thenb=aeforsomea2A,andhence,be=aee=ae2=ae=b.Hence,sincee2Aandf(e)2Ae,wehavethat(f)=ef(e)e=f(e2)e=f(e)e=f(e)2eAeforallf2E.Itisclearthatisavectorspacehomomorphism.Now,letf,g2E.Then(fg)=f(g(e)),and(f)op(g)=(g)(f)=g(e)f(e)=f(g(e))sinceg(e)2AandfisanA-endomorphism.Also,since(IdAe)=e,whichistheidentityofeAe,itfollowsthatisanF-algebrahomomorphism.Now,iff2ker(),then0=f(e)=af(e)=f(ae)foralla2A,whichimpliesthatfisthezeromaponAe,soisinjective.Now,wewillshowthatissurjective.Letx2eAe.Thenx=eweforsomew2A.Hence,wemaydeneamapx:Ae!Aebyx(s)=sx2Aeforalls2Ae.Wewillshowthatx2E.Let2F,s,t2Ae.Thenx(s+t)=(s+t)x=sx+tx=x(s)+x(t), 15

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soxisavectorspacehomomorphism.Furthermore,ifa2A,thenx(as)=asx=ax(s),sox2E.Now,(x)=x(e)=ex=eewe=ewe=x.Hence,issurjectiveandtheproofiscomplete. Corollary2.4. LetAAbetheleftA-moduleA.ThenA'EndA(AA)op.asF-algebras. Proof. Sete=1andapplyLemma 2.3 Now,recallthedenitionofbasicalgebraofAasin[ 1 ,p.23].LetAAbetheregularleftA-moduleA.WemaywriteAA=nMi=1Ai,whereA1,...,AnareprincipalindecomposablemodulesofA.ItmayhappenthatsomeoftheAiareisomorphic.ReordertheAiifnecessarysothatA1,...Ar,rn,isacompletesetofnonisomorphicprincipalindecomposablemodules.ThentheF-algebraEndA(A1Ar)opisthebasicalgebraofA.Notethatthebasicalgebraisdeneduptoisomorphism,sowemay,attimes,refertoabasicalgebraofA.TwoalgebrasaresaidtobeMoritaequivalentiftheyhaveisomorphicbasicalgebras.AtheoremofMoritain[ 16 ]yieldsthatthebasicalgebraofAisaninvariantofthecategoryofA-modules. Theorem2.5(Morita). LetAandBbetwoF-algebras.LetAModandBModbethecategoriesofleftA-andB-modules,respectively.ThenAModandBModareequivalentasabeliancategoriesifandonlyifAandBareMoritaequivalent.Furthermore,AandMat(n,A)areMoritaequivalent. 16

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Awell-knownfeatureofthebasicalgebraisthatallofitsirreduciblerepresentationsarelinear.Weprovethisfactbyconsideringidempotentsbelow. Theorem2.6. LetAbeanitedimensionalF-algebra.ThenthebasicalgebraofAhasthesamenumberofirreduciblerepresentationsasA,andeveryirreduciblerepresentationofthebasicalgebraofAislinear. Proof. LetA1,...,ArbeacompletesetofnonisomorphicprincipalindecomposableA-modules.By[ 6 ,Corollary(15.13)]and[ 6 ,Corollary(15.14)],Ai7!Top(Ai)isabijectionfromfA1,...,ArgtothesetofisomorphismclassesofirreduciblerepresentationsofA.Inparticular,thenumberofirreduciblerepresentationsofAisr.Now,sinceA1,...,ArarePIMs,thereexistsasetofprimitiveorthogonalidempotentse1,...,ersuchthatAi=Aeiforalli.Hence,ifwesete=e1++er,thenE=eAeisabasicalgebraofA.ItsufcestoshowthatthesemisimplealgebraE=J(E)isisomorphictoadirectsumofrcopiesofF.Notethatfe1=ee1e,...,er=eeregEisasetofprimitiveorthogonalidempotentsofEwhosesumise,theidentityofE.Hence,byLemma 2.2 ,E=Ee1Eerasadirectsumofprimitiveindecomposablemodules.But,Eei=eAeei=eAei.Sincefe1,...,ergisasetoforthogonalidempotents,wehavethatEei=e1AeierAeiasavectorspaceforeach1ir.Now,weclaimthatifi6=j,theneiAejJ(E).Indeed,sinceeiej=0,thecosetsei+J(A)andej+J(A)liveintwodifferentminimaltwo-sidedideals,sayIandJ,respectively,ofthesemisimplealgebraA=J(A).ButtheneiAej+J(A)I\J,sothateiAejJ(A).Now,by[ 6 ,Theorem(54.6)],J(E)=eJ(A)e.Hence,sinceeiAejJ(A),sothateiAej=eeiAejeJ(E).Now,sinceE=rXi,j=1eiAej,andJ(E)=eJ(A)eisanF-algebrainitsownright,wehavethatE=(e1Ae1erAer)+J(E) 17

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asasumofF-algebras.Now,weclaimthateiAei\J(E)=eiJ(A)ei.Indeed,ifeixei2eiJ(A)eiforx2J(A)A,theneixei=eeixeie2eJ(A)e=J(E),sothateiJ(A)eieiAei\J(E).Ify2eiAei\J(E),theny=eixei=ezeforsomex2A,z2J(A),thenbymultiplyingonbothsidesbyei,wegety=eixei=eizei2eiJ(A)ei,sothateiAei\J(E)=eiJ(A)ei.Thus,E=J(E)=((e1Ae1erAer)+J(E))=J(E)'e1Ae1=e1J(A)e1erAer=erJ(A)er.Now,by[ 6 ,Theorem(54.9)]and[ 6 ,Lemma(54.8)],thealgebraeiAei=J(eiAei)isaskeweld,butasFisanalgebraicallyclosedeld,thisimpliesthateiAei=J(eiAei)'FasanF-algebra.Finally,by[ 6 ,Theorem(54.6)]again,J(eiAei)=eiJ(A)ei,andhence,bytheabove,E=J(E)isisomorphictothedirectsumofrcopiesofFasanF-algebra,completingtheproofofthetheorem. Theorem2.7. SupposeAisanitedimensionalF-algebraandthateveryirreduciblerepresentationofAisone-dimensional.ThenAisisomorphictoitsbasicalgebra. Proof. SinceeveryirreduciblerepresentationofAisone-dimensional,thesemisimplealgebraA=J(A)isisomorphictoFr,whereristhenumberofirreduciblerepresentationsofA.Leteibetheidentityofthei-thdirectsummandofFr.Theneiisprimitive,eiej=0ifi6=j,ande1++eristheidentityofA=J(A).Hence,by[ 1 ,Corollary1.7.4],thereexistprimitiveorthogonalidempotentsf1,...,fr2Asuchthat1=f1++fr.Hence,byLemma 2.2 ,A=rMi=1Afi.By[ 6 ,Corollary(54.13)]and[ 6 ,Corollary(54.13)],themapAfi7!Top(Afi)isasurjectionfromthesetofisomorphismclassesofprincipalindecomposableA-modulestothesetofisomorphismclassesofirreduciblerepresentations.SinceAhasrirreduciblerepresentations,thismapisabijection.Hence,AfiandAfjarenonisomorphicifi6=j. 18

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Finally,thebasicalgebraofAisEndA rMi=1Afi!op=EndA(A)op'AbyCorollary 2.4 LetfA1,...,ArgbeacompletesetofnonisomorphicprincipalindecomposableA-modules.Then,byLemma 2.1 ,Ai=Aeiforsomeprimitiveorthogonalidempotentse1,...,er2A.Now,notethatsincetheeiareorthogonalidempotents,wehavethatnMi=1Ai=rMi=1Aei=A(e1++er).Hence,byLemma 2.3 ,thebasicalgebraofAisisomorphictoeAe,wheree=e1++er.Notethatsincetheeiareorthogonal,eisanidempotent.Anidempotente2AofthisformiscalledabasicidempotentofA.Theorems 2.6 and 2.7 implythatAisisomorphictoitsbasicalgebraifandonlyifeveryirreduciblerepresentationofAislinear.Inthiscase,wesaythatthealgebraAisbasic. Theorem2.8. LetAbeanitedimensionalF-algebra,andlete2Abeabasicidempotent.ThenAisbasicifandonlyife=1,theidentityofA. Proof. WehavethateAeA.Also,eAeisanalgebrawithidentitye.IfAisbasic,theneAeandAhavethesamedimensionasF-vectorspaces.Inparticular,eAe=A.SinceAisanalgebrawithidentity1,itmustbethate=1.Conversely,ife=1,theneAe=A. Theorem2.9. SupposeAisanalgebraoveranalgebraicallyclosedeldandthatAhasaunique(uptoisomorphism)irreduciblerepresentation.ThenAisisomorphictothealgebraofn-by-nmatricesoverthebasicalgebraofA,wherenisthedegreeoftheuniqueirreduciblerepresentationofA. 19

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Proof. LetAbetheleftA-moduleA.Then,sinceAhasauniqueirreduciblerepresentation,wemaywriteA'P...P=mPwherePisaprojectivecoveroftheuniqueirreducibleA-module.ByCorollary 2.4 ,A'EndA(A)op.Hence,A'EndA(A)op'EndA(mP)op'(EndA(P)FMat(m,F))op'EndA(P)opFMat(m,F).Hence,sinceEndA(P)opisthebasicalgebraofA,Aisisomorphictom-by-mmatricesoveritsbasicalgebra.SincetheuniqueirreduciblerepresentationofAcanbeviewedasanirreduciblerepresentationofthebasicalgebra(allofwhicharelinearby 2.6 )tensoredwithanirreduciblerepresentationofMat(m,F)(whichhasdegreem),wehavethatm=nandwearedone. 2.2CharactersandBlocksofFiniteGroupsInthissection,werecallsomebasicfactsaboutcharactersandblocksofp-solvablegroups.Throughout,letGbeanitegroup.Fixaprimenumberp.WerstrecallthedenitionofanordinarycharacterofG.Let:G!GLn(C),n0,beaC-representation(whichisalsocalledanordinaryrepresentation)ofG.Recallthatthecharacteraffordedbyisthemap:G!Cdenedby(g)=Tr((g)),whereTristhetracefunction.Thedegreeofistheintegern.Wesaythatacharacterisirreducibleiftherepresentationwhichaffordsisirreducible.WewriteIrr(G)forthesetofirreduciblecharactersofG.Astandardtextonthecharactertheoryofnitegroupsis[ 12 ].ByMaschke'sTheorem(see,forexample,[ 17 ,Theorem(1.21)]),thegroupalgebraCGissemisimpleandis,hence,byWedderburn'sTheorem[ 17 ,Theorem(1.17)],CGisadirectsumoffullmatrixalgebras.Eachcharacter2Irr(G)correspondsuniquelyto 20

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oneofthesefullmatrixalgebrasinthefollowingway.For2Irr(G),letf=(1) jGjXg2G (g)g2Z(CG),theprimitiveidempotentofZ(CG)associatedto.ThenCGfisafullmatrixalgebraoverCofdegree(1),andasaCG-module,CGfisisomorphictothedirectsumof(1)copiesofM,whereMistheirreducibleCG-modulewhichaffords.WewillnowrecallthedenitionofaBrauer(ormodular)characterofG,whicharethecharacteristicpanalogsofordinarycharacters,asin[ 17 ].First,wewilldeneaparticularalgebraicallyclosedeldofcharacteristicp.LetRbetheringofalgebraicintegersinC.ChooseamaximalidealMofRwhichcontainstheintegerp.ThenR=Misaeldofcharacteristicp.LetF=R=M.Dene:R!Ftobethenaturalsurjectiveringhomomorphismgivenbyr=r+Mforallr2R.Wecallreductionmodulop.ThefollowinglemmashowsthatFistheappropriateeldtodeneBrauercharacters. Lemma2.10. LetU2Cbethegroupofp0-rootsofunity.Thatis,Uisthemultiplicativegroupofcomplexrootsof1whoseordersarenotdivisiblebyp.ThentherestrictionoftoUdenesanisomorphism:U!Fofmultiplicativegroups.Furthermore,FisthealgebraicclosureofitsprimeeldZ'Zp.LetG0bethesetofp-regularelementsofG.Thatis,G0isthesetofelementsofGwhoseordersarenotdivisiblebyp.Let:G!GLn(F)beanF-representationofG.Ifg2G0,then(countingmultiplicity)(g)hasneigenvalues,allofwhichareinF.Hence,byLemma 2.10 ,theeigenvaluesof(g)canbewrittenas1,...,n2Fforuniquelydetermined1,...,n2U.WenowsaythattheBrauercharacter(ormodularcharacter)affordedbyisthemap':G0!Cdenedby'(g)=1++nasabove.WesaythataBrauercharacter'isirreducibleitisaffordedbyanirreduciblerepresentation.DenotethesetofirreducibleBrauercharactersbyIBrp(G).Throughout,sincetheprimepisxed,wewillsimplywriteIBr(G). 21

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Blocktheory(forwhichastandardtextis[ 17 ])isatoolforrelatingtheordinaryandBrauercharactersofanitegroup.Sincethesecharacterscarryinformationabouttheircorrespondingrepresentations,blocktheoryisalsousedtostudytherelationshipsbetweentheordinaryandmodularrepresentationsofG(equivalently,CG-modulesandFG-modules).Wewilldenethep-blocksofGasin[ 17 ].For2Irr(G),let0=jG0,therestrictionoftoG0.By[ 17 ,Corollary(2.9)],0isaBrauercharacterofG.Hence,thereexistintegersd'0indexedby'2IBr(G)suchthat0=X'2IBr(G)d''.Theintegersd',2Irr(G),'2IBr(G)arecalledthedecompositionnumbers.WecannowdenetheblocksofG. Denition2.11. Wesaythattwoordinarycharacters, 2Irr(G)arelinkedifthereexistssome'2IBr(G)suchthatd'6=06=d '.TheBrauergraphofGisthegraphwhoseverticesaretheelementsofIrr(G)andadjacencyisdenedbylinkage.AblockBofGisasubsetofIrr(G)[IBr(G)containingsubsetsIrr(B)Irr(G)andIBr(B)IBr(G)suchthat: Irr(B)isaconnectedcomponentoftheBrauergraph. IBr(B)=f'2IBr(G)jd'>0forsome2Irr(B)g. B=Irr(B)[IBr(B).WedenotethesetofblocksofGbyBl(G).EachblockofGcorrespondstoanindecomposabletwo-sidedidealofthegroupalgebraFG.IfB2Bl(G),thentheblockidempotentofB,eBisgivenbyeB=X2Irr(G)e, 22

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wheree=f=0@(1) jGjXg2G (g)g1A2Z(FG).TheneacheBisaprimitiveidempotentofZ(FG),FGeBisanindecomposabletwo-sidedidealofFG,andFG=XB2Bl(G)FGeB.GivenB2Bl(G),FGeBistheblockalgebraofB.Cliffordtheoryisthestudyoftherelationshipsbetweentherepresentationtheoryofagroupandanormalsubgroup.SuchrelationshipswereoriginallystudiedbyAlfredCliffordin[ 5 ].IfNisanormalsubgroupofG,thenthereisanactionofGonIrr(N)(respectivelyIBr(N))givenbyg(x)=(gxg)]TJ /F9 7.97 Tf 6.58 0 Td[(1)forall2Irr(N)(respectively2IBr(N)).If2Irr(N)[IBr(N),wedenotebyIG()thestabilizerinGofthisactionon.WecallIG()theinertiagroupofinG.WewillnowsummarizesomeresultsfromCliffordtheory.WewillrststatetheordinaryandmodularversionsofClifford'stheoremandCliffordcorrespondence.Theordinaryversionsarein[ 12 ,Theorem(6.2)]and[ 12 ,Theorem(6.11)],repectively.Themodularversionsarein[ 17 ,Corollary(8.7)]and[ 17 ,Theorem(8.9)],respectively. Theorem2.12(Clifford'sTheorem,OrdinaryVersion). LetNGandlet2Irr(G)andlet2Irr(G).LetbeanirreducibleconstituentofNandsuppose=1,...,tarethedistinctconjugatesofinG.ThenN=etXi=1iwheree=[N,]. Theorem2.13(CliffordCorrespondence,OrdinaryVersion). LetNG,let2Irr(N),andT=IG().Then 7! GisabijectionfromIrr(Tj)ontoIrr(Gj). Theorem2.14(Clifford'sTheorem,ModularVersion). LetNG.Let'2IBr(G)andlet2IBr(N).Then'isanirreducibleconstituentofGifandonlyifisanirreducible 23

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constituentof'N.Inthiscase,if=1,...,tarethedistinctG-conjugatesof,then'N=etXi=1i. Theorem2.15(CliffordCorrespondence,ModularVersion). LetNG,let2IBr(N),andT=IG().Then 7! GisabijectionfromIBr(Tj)ontoIBr(Gj).WecanrenetheresultofTheorem 2.14 inthecasethatG=Nisap-group.ThisisthecontentofGreen'sTheorem[ 17 ,Theorem(8.11)]. Theorem2.16(Green). SupposethatG=Nisap-groupandlet2IBr(N).Thenthereexistsaunique'2IBr(Gj).Furthermore,'NisthesumofthedistinctG-conjugatesof.Inparticular,ifisG-invariant,then'N=.TheactionofGonIrr(N)andIBr(N)alsopreservestheblockstrucureofG.Moreprecisely,ifb2Bl(N),thenthesetbg=fgj2Irr(b)[IBr(b)gisalsoablockofG.Thus,GactsonBl(N)byconjugation.WewishtostateaCliffordtheoremforblocks,butrstwemustdiscussblockcoverings.GivenB2Bl(G)andb2Bl(N),wesaythatBcoversbifthereexistblocksB=B1,...,Bs2Bl(G)andb=b1,...,bt2Bl(N)suchthatsXi=1eBi=tXi=1ebs.Anequivalentdenitionofblockcoveringcanbefoundin[ 17 ,Theorem(9.2)],whichwestatenow. Theorem2.17. LetNbeanormalsubgroupofG.Letb2Bl(N)andletB2Bl(G).Thenthefollowingareequivalent: Bcoversb. If2B,theneveryirreducibleconstituentofNliesinaG-conjugateofb. Thereisa2BsuchthatNhasanirreducibleconstituentinb. 24

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NotethatTheorem 2.17 impliesthatifb1,b2arebothcoveredbyB,thenb1andb2areG-conjugate.WedenotethesetofblockswhichcoverbbyBl(Gjb).WeconcludethissectionwiththeFong-ReynoldsTheorem[ 17 ,Theorem(9.14)],whichisananalogoftheCliffordtheoremsforblocks. Theorem2.18. LetNbeanormalsubgroupofGandletb2Bl(N).LetT(b)bethestabilizerofbinG. ThemapBl(T(b)jb)toBl(Gjb)givenbyB7!BGisawell-denedbijection. IfB2Bl(T(b)jb),thenIrr(BG)=f Gj 2Irr(B)gandIBr(BG)=f'Gj'2IBr(B)g. IfB2Bl(T(b)jb),theneverydefectgroupofBisadefectgroupofBG. 2.3CharactersandBlocksofp-SolvableGroupsLetGbeanitegroup,andletbeasetofprimes.AHall-subgroupofGisasubgroupHofGsuchthatjHjisdivisibleonlybyprimesinandjG:Hjisnotdivisiblebyanyprimesin.Hence,forqaprime,theHallfqg-subgroupsofGarepreciselytheSylowq-subgroups.RecallthatGissolvableifeverycompositionfactorofGiscyclicofprimeorder.PhillipHall'stheorem,whichappearsin[ 13 ,Theorem6.4.10]characterizessolvablegroupsintermsofHallsubgroups. Theorem2.19(P.Hall). LetGbeanitegroup.Thenthefollowingareequivalent: Gissolvable. GcontainsaHall-subgroupforeverysetofprimes. GcontainsaHallp0-subgroupforeveryprimep.AHallp0-subgroupisalsocalledap-complement.RecallthatagroupGisp-solvableifeverynonabeliancompositionfactorofGisap0-group.From[ 13 ,Theorem6.4.6],itfollowsthatap-solvablegrouphasap-complementandthateveryp-complementisconjugateinG.Wewillnowdiscussthecharactersandblocksofp-solvablegroups.RecallthatanitegroupGisp-solvableifeverynonabeliancompositionfactorofGisap0-group. 25

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Thecharactersandblocksofp-solvablegroupswereoriginallystudiedbyPaulFongin[ 7 ].Themainresultsinthecharactertheoryandblocktheoryofp-solvablegroups(whicharesurveyedin[ 17 ,Chapter10])reducethemodularrepresentationtheorytoordinaryrepresentationtheoryinsomeway.TherstsuchresultwewilllististhefamedFong-Swantheorem,whichappearsin[ 17 ,Theorem(10.1)]. Theorem2.20(Fong-Swan). SupposethatGisp-solvable.If'2IBr(G),then'=0forsome2Irr(G).TheFong-SwantheoremallowsustocalculatetheirreducibleBrauercharactersofap-solvablegroupcompletelyintermsoftheordinarycharacters,asdemonstratedinthefollowingcorollary,foundin[ 17 ,Corollary(10.4)]. Corollary2.21. SupposeGisp-solvable.ThenIBr(G)isthesetofall0suchthat2Irr(G)and0isnotoftheform0+0fornonzeroordinarycharacters,ofG.Let'2IBr(G),whereGisp-solvable.Aliftof'isacharacter2Irr(G)suchthat'=0.TheremaybemanyliftsofagivenirreducibleBrauercharacter'.AtheoremofIsaacs,whichappearsin[ 17 ,Theorem(10.6)],statesthat(forpodd)thereissubsetofIrr(G)suchthatliftingtothissubsetisinjective.Recallthatacharacter2Irr(G)isp-rationalifitsvaluesareinQ(jGjp0),theeldobtainedbyadjoiningtoQaprimitivep0-rootofunity. Theorem2.22(Isaacs). SupposethatGisp-solvable,andpisodd.If'2IBr(G),thenthereexistsauniquep-rationalliftof'.Sincep-solvablegroupshavep-complements,wecandiscusstherelationshipsbetweenthecharactersofap-solvablegroupGandthoseofap-complementHofG.RecallthatsincepdoesnotdividetheorderofH,IBr(H)=Irr(H).Thefollowingtheorem,foundin[ 17 ,Theorem(10.9)],showswhathappenswhenirreducibleBrauercharactersofGarerestrictedtoH. Theorem2.23. SupposethatGisp-solvableandletHbeap-complementofG.Thenthemap'7!'Hfromf'2IBr(G)jp-'(1)g!Irr(H)isawell-denedinjectivemap. 26

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LetMbeanirreducibleFG-modulewithprojectivecoverP.IfMaffords'2IBr(G),thenrecallthatPaffordstheBrauercharacter'=X2Irr(G)d'0.TheBrauercharacter'iscalledtheprojectiveindecomposablecharacterof'.AtheoremofFongin[ 17 ,Theorem(10.13)]statesthattheprojectiveindecomposablecharactersareinducedfromcertainirreduciblecharactersofp-complementsubgroups. Theorem2.24(Fong). SupposethatGisp-solvableandletHbeap-complementofG.If'2IBr(G),thenthereexistsacharacterofHsuchthatG='.Furthermore,everysuchisirreducibleandhasdegree'(1)p0. Corollary2.25(Fong'sDimensionFormula). IfGisp-solvableand'2IBr(G),then'(1)=jGjp'(1)p0.Acharacterofap-complementHsuchthatG='issaidtobeaFongcharac-terfor'.Thenexttheorem,[ 17 ,Theorem(10.18)],identiestheFongcharacters. Theorem2.26. SupposeGisp-solvableandHisap-complementofG.If'2IBr(G),thentheFongcharactersof'aretheirreducibleconstituentsof'Hofsmallestpossibledegree.Thisdegreeis'(1)p0.Toconcludethissection,westateatheoremofFong,foundin[ 17 ,Theorem(10.20)],which,togetherwhichthetheoryofblockcoveringandTheorem 2.18 isthemaintoolforstudyingblocksofp-solvablegroups. Theorem2.27(Fong). SupposeGisp-solvableandlet2Irr(Op0(G))beG-invariant.Thenthereisauniquep-blockofGcoveringtheblockfgofOp0(G).Also,Irr(B)=Irr(Gj)andIBr(B)=IBr(Gj).Furthermore,thedefectgroupsofBaretheSylowp-subgroupsofG.Finally,westateacorollarytoTheorem 2.27 ,asfoundin[ 11 ],whichshowsthatthehypothesisthatisG-invariantisnottoorestrictive. 27

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Corollary2.28(Isaacs). LetGbep-solvable,andletB2Bl(G).ThenthereexistsasubgroupJofGandacharacter 2Irr(M),whereM=Op0(J),suchthatinductiondenesbijectionsfromIrr(Jj )ontoIrr(B)andfromIBr(Jj )ontoIBr(B). 28

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CHAPTER3BLOCKIDEMPOTENTSFORP-SOLVABLEGROUPSInthissection,wediscusstheblockidempotentseBforblocksBofp-solvablegroups.WeproceedusingthenotationofChapter2. 3.1BlocksofFullDefectIfBisablockofanitegroupG,thenBofsaidtobeoffulldefect(orhasfulldefectifthedefectgroupsofBarepreciselytheSylowp-subgroupsofG.RecallthatBhasfulldefectifandonlyifBcontainsanirreducibleordinarycharacterwhosedegreeisnotdivisiblebyp.By[ 17 ,Corollary(3.17)],anotherequivalentconditionforBtohavefulldefectisthatBcontainsanirreducibleBrauercharacterwhosedegreeisnotdivisiblebyp.Tobegin,weprovideanEnglishtranslationofatheoremwhichappearsinGermanin[ 19 ,Satz2.1]. Theorem3.1(Tiedt). LetBbeablockofap-solvablegroupG.IfadefectgroupofBcontainsaSylowp-subgroupofOp0p(G),thenthesupportofeBiscontainedinOp0(G).TheproofofTheorem 3.1 reliesonseveraldeeperresults,duetoKnorr,Harris,andOsima,presentedhereaslemmas.Theseresultscanbefoundin[ 17 ,Theorem(9.26)],[ 9 ,Corollary2],and[ 17 ,Corollary(3.8)],respectively. Lemma3.2(Knorr'sTheorem). LetGbeanitegroup.Letb2Bl(NjQ),whereN/G.IfB2Bl(G)coversb,thenthereisadefectgroupPofBsuchthatP\N=Q. Lemma3.3(Harris-Knorr). LetGbeanitegroup.LetN/Gandb2Bl(NjQ).IfCG(Q)N,thenbGisdenedandistheonlyblockofGwhichcoversb. Lemma3.4(Osima'sTheorem). LetGbeanitegroup.IfB2Bl(G),theneBhassupportinG0.WecannowproveTheorem 3.1 ProofofTheorem 3.1 SupposeGisaminimalcounterexample.SupposeG=Op0p(G).ThenGisp-nilpotentandhenceG0=Op0(G).Then,byLemma 3.4 ,thesupportof 29

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eBiscontainedinOp0(G),acontradiction.Hence,Op0p(G)isapropersubgroupofG.Considertheascendingp-series1Op0(G)Op0p(G)Op0pp0(G)G.LetHbethelargestpropersubgoupofGintheaboveseries.ThenH/GandOp0p(G)H.SinceHisnormalinGandOp0(G)H,Op0(H)=Op0(G).Similarly,Op0p(H)=Op0p(G).Now,by[ 17 ,Corollary(9.3)],BcoversexactlyoneG-conjugacyclassofblocksfb1,...,bngofH.Then,bydenitionofblockcovering,n0Xi=1eBi=nXj=1ebi,wherefB=B1,B2,...,Bn0gisthecollectionofblocksofGwhichcoverfb1,...,bng.Itsufcestoshowthatn0=1andthateachebjhassupportinOp0(H).Foreach1jn,letQjbeadefectgroupofbj.Then,byLemma 3.2 ,thereexistdefectgroupsPjofBsuchthatQj=Pj\Hforallj.LetSjbeaSylowp-subgroupofOp0p(G)suchthatSjPj.Then,sinceSjOp0p(G)HandSjPj,itholdsthatSjQj=Pj\H.Now,sinceQjcontainsaSylowsubgroupofOp0p(G)andOp0p(H)=Op0p(G),QjcontainsaSylowsubgroupofOp0p(H).Byinduction,eachebjhassupportinOp0(H).Finally,sinceGisp-constrained,wehavethatCG(S1)Op0p(G).Hence,CG(Q1)CG(S1)Op0p(G)H.Hence,byLemma 3.3 ,BistheonlyblockofGthatcoversb1.Thus,eB=nXj=1ebjandwearedone. 30

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Corollary3.5(Tiedt). LetBbeablockofap-solvablegroupG.IfBhasfulldefect,thenthesupportofeBiscontainedinOp0(G). Proof. SinceBhasfulldefect,thedefectgroupsofBarepreciselytheSylowp-subgroupsofG.Hence,anySylowsubgroupofOp0p(G)iscontainedinsomedefectgroupofB,andwemayapplyTheorem 3.1 WewillnowprovidesomealternatewaysofcalculatingtheblockidempotenteBofablockBofap-solvablegroup.Theorem 3.6 providesacalculationofeBintermsofirreduciblecharactersofOp0(G)and 3.8 providesacalculationintermsoftheirreducibleBrauercharactersofB. Theorem3.6. LetGbeap-solvablegroupandB2Bl(G).IfBhasfulldefect,thenthereexistsauniqueG-conjugacyclassofcharactersf1,...,tgIrr(Op0(G))suchthatIG(1)containsaSylowp-subgroupofG,andeB=tXi=1ei. Proof. Bcoversauniqueconjugacyclassofp-blocksofOp0(G).Sincethep-blocksofOp0(G)arepreciselyffgj2Irr(Op0(G))g,wehavethatthereexistsauniqueG-conjugacyclassf1,...,tgIrr(Op0(G))(wheret=jG:IG(1)j)andpositiveintegersm,2BsuchthatjOp0(G)=mtXi=1iforall2B.Notethateachihasthesamedegree.SinceBhasfulldefect,thereexistsacharacter2Bsuchthatp-(1).But(1)=mtXi=1i(1)=mt1(1)=mjG:IG(1)j1(1).Inparticular,p-jG:IG(1)j.Hence,IG(1)containsaSylowp-subgroupofG.Now,let =tXi=1i,sothatjOp0(G)=m forall2B.TheneB=0@1 jGjXg2GX2Irr(B)(1)(g)]TJ /F9 7.97 Tf 6.58 0 Td[(1)g1A. 31

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But,byCorollary 3.5 ,eBhassupportcontainedinOp0(G).Hence,eB=0@1 jGjXg2Op0(G)X2Irr(B)(1)(g)]TJ /F9 7.97 Tf 6.59 0 Td[(1)g1A=0@1 jGjXg2Op0(G)X2Irr(B)m (1)m (g)]TJ /F9 7.97 Tf 6.59 0 Td[(1)g1A=0@1 jGjXg2Op0(G) (1) (g)]TJ /F9 7.97 Tf 6.59 0 Td[(1)gX2Irr(B)m21A=0@ (1)M jGjXg2Op0(G) (g)]TJ /F9 7.97 Tf 6.59 0 Td[(1)g1A2Z(FG)WhereMistheconstantX2Irr(B)m2.Ontheotherhand,tXi=1ei=tXi=10@1 jOp0(G)jXg2Op0(G)i(1)i(g)]TJ /F9 7.97 Tf 6.59 0 Td[(1)g1A=0@1(1) jOp0(G)jXg2Op0(G)tXi=1i(g)]TJ /F9 7.97 Tf 6.59 0 Td[(1)g1A=0@1(1) jOp0(G)jXg2Op0(G) (g)]TJ /F9 7.97 Tf 6.59 0 Td[(1)g1A2Z(FOp0(G)).Sincetheeiej=ijei,wehavethattXi=1eiisanidempotentofZ(FOp0(G))thatdiffersfromeBonlybyanelementofF.Hence,sinceeB6=0,eB=tXi=1ei. 32

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Lemma3.7. LetGbeap-solvablegroup,andsupposethatB2Bl(G)hasfulldefect.TheneB=0@1 jGjp0Xg2Op0(G)X'2IBr(B)'(1)p0 '(g)g1A2Z(FG). Proof. By[ 17 ,Page55],eB=0@1 jGjXg2GX2Irr(B)(1) (g)g1A2Z(FG).Then,byapplying[ 17 ,3.8],[ 17 ,3.6],and[ 17 ,10.14],weobtaineB=0@1 jGjXg2GX2Irr(B)(1) (g)g1A=0@1 jGjXg2G0X2Irr(B)(1) (g)g1A=0@1 jGjXg2G0X'2IBr(B)'(1) '(g)g1A=0@1 jGjXg2G0X'2IBr(B)jGjp'(1)p0 '(g)g1A=0@jGjp jGjXg2G0X'2IBr(B)'(1)p0 '(g)g1A=0@1 jGjp0Xg2Op0(G)X'2IBr(B)'(1)p0 '(g)g1A. Corollary3.8. LetGbeap-solvablegroup.SupposeB2Bl(G)hasfulldefectandthat'(1)p01modulopforall'2IBr(B).TheneB=1 jGjp0Xg2Op0(G)X'2IBr(B) '(g)g2Z(FG). 33

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Inparticular,forallg2G0nOp0(G),wehaveX'2IBr(B)'(g)=02F. Proof. ByLemma 3.7 ,eB=0@1 jGjp0Xg2Op0(G)X'2IBr(B)'(1)p0 '(g)g1A2Z(FG)=1 jGjp0Xg2Op0(G)X'2IBr(B) '(g)g2Z(FG).Hence,ifg2G0nOp0(G),thecoefcientofgineBis0.Butthiscoefcientis1 jGjp0X'2IBr(B) '(g). 3.2BlocksContainingaLinearBrauerCharacterWecansaymoreabouttheblockidempotentofablockwhichcontainsalinearBrauercharacter.Infact,itcanberealizedasthecharacteridempotent(inFG)ofalinearcharacterofOp0(G),andwecanndthecorrespondingcharacter. Lemma3.9. LetBbeablockofap-solvablegroupG.IfBcontainsalinearBrauercharacter,thenthereexistsauniqueB2Irr(Op0(G))suchthat'jOp0(G)='(1)Bforall'2IBr(B).Furthermore,B(1)=1,BisinvariantundertheactionbyconjugationofGonIrr(Op0(G)),Irr(B)=Irr(GjB),andIBr(B)=IBr(GjB). Proof. Suppose 2IBr(B)islinear.Then jOp0(G)isalinearirreduciblecharacter2Irr(Op0(G)).SetB=.ItisapparentthatB(1)=1.By[ 17 ,Corollary(8.7)],BisG-invariant.Hence,theblockfBg2Bl(Op0(G))isG-invariant.By[ 17 ,Theorem(9.2)],if'2IBr(B),theneveryirreducibleconstituentof'jOp0(G)liesinaG-conjugateoftheblockfBg.Hence,Bistheonlyirreducibleconstituentof'jOp0(G).Thisimpliesthat'isamultipleofB.Hence,'jOp0(G)='(1)Bforall'2IBr(B).ThefactthatIrr(B)=Irr(GjB)andIBr(B)=IBr(GjB)followsfrom[ 17 ,Theorem(10.20)]. 34

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Denition3.10. ForBablockofap-solvablegroupGsuchthatBcontainsalinearBrauercharacter,deneBtobetheuniqueirreduciblecharacterofOp0(G)givenbyLemma 3.9 above. Theorem3.11. LetBbeablockofap-solvablegroupG.IfBcontainsalinearBrauercharacter,theneB=eB. Proof. By[ 17 ,Corollary(9.3)],BcoversauniqueG-conjugacyclassofblocksofOp0(G).SincetheblockfBgisG-invariant,thismeansthateB=tXi=1eBiwherefB=B1,...,BtgisthesetofblocksofGwhichcoverfBg.But,by[ 17 ,Theorem(10.20)],BistheuniqueblockofGwhichcoversfBg.HenceeB=eB. Inparticular,Theorem 3.11 impliesthatifBcontainsalinearBrauercharacter,then,sinceeBistheidempotentaffordedbyalinearcharacterofOp0(G),thesupportofeBinGisexactlyOp0(G). Corollary3.12. LetBbeablockofap-solvablegroupG.IfBcontainsalinearBrauercharacter,thenX'2IBr(B)'(g)6=0ifandonlyifg2Op0(G).Inparticular,ifGisap-solvablegroupandB0istheprincipalblockofB,thenOp0(G)=8<:g2G0:X'2IBr(B0'(g)6=09=;. Proof. Letg2G0.ByTheorem 3.8 ,X'2IBr(B)'(g)=0ifg=2Op0(G).Nowsupposeg2Op0(G).BytheproofofTheorem 3.8 ,thecoefcientofGineBisX'2IBr(B)'(g).SinceeB=eBandBisalinearcharacter,thecoefcientofgineBisnonzero.Hence,X'2IBr(B)'(g)6=0ifg2Op0(G). 35

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Theorem3.13. LetGbeap-solvablegroup,andsupposeB2Bl(G)containsalinearBrauercharacter.ThentheblockalgebraFGeBisisomorphictotheprincipalblockalgebraF[G=Op0(G)].Inparticular,dimF(FGeB)=jG:Op0(G)j. Proof. Let'2IBr(B)belinear.Let2Irr(G)besuchthat0='.SuchacharacterexistsbytheFong-SwanTheorem[ 17 ,Theorem(10.1)].ThenbyLemma 3.9 ,jOp0(G)='jOp0(G)=B.Considerthefunction0:G!F[G=Op0(G)]givenby0(g)=(g)gOp0(G)forallg2G.Then0extendstoasurjectivevectorspacehomomorphism1:FG!F[G=Op0(G)]intheobviousway.Furthermore,sinceislinear,1(g)1(h)=((g)gOp0(G))((h)hOp0(G))=(gh)ghOp0(G)=1(gh)forallg,h2G.Hence,1isasurjectivehomomorphismofF-algebras.Now,notethatbyTheorem 3.11 ,1(eB)=1(eB)=10@1 jOp0(G)jXg2Op0(G)B(g)]TJ /F9 7.97 Tf 6.58 0 Td[(1)g1A=1 jOp0(G)jXg2Op0(G)B(g)]TJ /F9 7.97 Tf 6.59 0 Td[(1)(g)gOp0(G)=1 jOp0(G)jXg2Op0(G)B(g)]TJ /F9 7.97 Tf 6.59 0 Td[(1)B(g)gOp0(G)=1 jOp0(G)jjOp0(G)jOp0(G)=Op0(G)2F[G=Op0(G)].Thus,1mapseBtotheidentityofthealgebraF[G=Op0(G)].So1(xeB)=1(x)forallx2FG.Thus,ifxeB=yeB2FGeB,then1(x)=1(y).Therefore,wemaydeneasurjectivealgebrahomomorphism:FGeB!F[G=Op0(G)]by(xeB)=1(x)forallx2FG. 36

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Itremainstoshowthatisinjective.Wewillshowtheexistenceofaleftinverseto.Considerthemap0:G=Op0(G)!FGeBgivenby0(gOp0(G))=(g)]TJ /F9 7.97 Tf 6.59 0 Td[(1)geB.Wemustshowthat0isawell-denedfunction.First,notethatifz2Op0(G),then(z)]TJ /F9 7.97 Tf 6.59 0 Td[(1)zeB=B(z)]TJ /F9 7.97 Tf 6.59 0 Td[(1)zeB=B(z)]TJ /F9 7.97 Tf 6.59 0 Td[(1)z1 jOp0(G)jXg2Op0(G)B(g)]TJ /F9 7.97 Tf 6.58 0 Td[(1)g=1 jOp0(G)jXg2Op0(G)B(g)]TJ /F9 7.97 Tf 6.58 0 Td[(1)B(z)]TJ /F9 7.97 Tf 6.58 0 Td[(1)zg=1 jOp0(G)jXg2Op0(G)B((zg))]TJ /F9 7.97 Tf 6.58 0 Td[(1)zg=eB=eB.Now,supposegOp0(G)=hOp0(G).Theng=hzforsomez2Op0(G).Hence,0(gOp0(G))=(g)]TJ /F9 7.97 Tf 6.59 0 Td[(1)geB=(z)]TJ /F9 7.97 Tf 6.58 0 Td[(1h)]TJ /F9 7.97 Tf 6.58 0 Td[(1)hzeB=(h)]TJ /F9 7.97 Tf 6.59 0 Td[(1)h(z)]TJ /F9 7.97 Tf 6.59 0 Td[(1)zeB=(h)]TJ /F9 7.97 Tf 6.59 0 Td[(1)heB=0(hOp0(G))bythepreviousobservation.Hence,0iswell-dened.Then,0maybeextendedlinearlytoavectorspacehomomorphism:F[G=Op0(G)]!FGeB.Now,ifg,h2G,then(gOp0(G))(hOp0(G))=(g)]TJ /F9 7.97 Tf 6.59 0 Td[(1)geB(h)]TJ /F9 7.97 Tf 6.59 0 Td[(1)heB=(h)]TJ /F9 7.97 Tf 6.58 0 Td[(1)(g)]TJ /F9 7.97 Tf 6.59 0 Td[(1)ghe2B=((gh))]TJ /F9 7.97 Tf 6.59 0 Td[(1)gheB=(ghOp0(G)). 37

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Hence,isanF-algebrahomomorphism.Finally,forallg2G,((geB))=(0(g))=((g)gOp0(G))=(g)(gOp0(G))=(g)(g)]TJ /F9 7.97 Tf 6.58 0 Td[(1)geB=geB.SinceFGeBisspannedbyfgeBjg2Gg,thisimpliesthatisaleftinverseto,andhenceisinjective. Theorem 3.13 impliesthatallofblocksofap-solvablegroupwhichcontainalinearcharacterareisomorphictotheprincipalblock.ThisimpliesthatanytwosuchblockscontaininingalinearBrauercharacterhavethesamenumberofBrauercharactersand(countingmultiplicities)thesameBrauercharacterdegrees.WeconcludethissectionwithabriefinvestigationofblockscontainingonlylinearirreducibleBrauercharacters.Thefollowinglemmaiswell-known,buttheproofisincludedhereforcompleteness. Lemma3.14. IfGisanitegroup,then\'2IBr(G)ker(')=Op(G). Proof. LetI=\'2IBr(G)ker(').By[ 17 ,Lemma(2.32)],Op(G)I.Clearly,IisanormalsubgroupofG.WewillshowthatIisap-subgroup.Letg2Ibeap-regularelement.Then,by[ 17 ,Lemma(6.11)],'(g)='(1)forall'2IBr(G).Hence,forall2Irr(G),wehavethat(g)=X'2IBr(G)d''(g)=X'2IBr(G)d''(1)=(1).Hence,g2\2Irr(G)ker()=f1g.Thus,g=1andIconsistsofp-elements.Hence,Iisanormalp-subgroup,soIOp(G). 38

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Theorem3.15. LetBbeablockofap-solvablegroupGandthateveryirreducibleBrauercharacterinBislinear.ThenthenumberofirreducibleBrauercharactersinB,isjG:Op0(G)jp0=jG:Op0p(G)j. Proof. ByTheorem 3.13 ,FGeBisisomorphictoF[G=Op0(G)].Hence,BhasthesamenumberofirreducibleBrauercharactersasG=Op0(G)andeveryirreducibleBrauercharacterofG=Op0(G)islinear.Hence,tocountthenumberofirreducibleBrauercharactersinB,itsufcestocountthenumberofirreducibleBrauercharactersofthegroupG=Op0(G).Now,sinceIBr(G=Op0(G))consistsoflinearBrauercharacters,wehavethat)]TJ /F2 11.955 Tf 5.48 -9.69 Td[(G=Op0(G)0\'2IBr(G=Op0(G))ker'=Op(G=Op0(G))=Op0p(G)=Op0(G)byLemma 3.14 .Hence,)]TJ /F2 11.955 Tf 5.48 -9.68 Td[(G=Op0(G)=)]TJ /F2 11.955 Tf 5.48 -9.68 Td[(Op0p(G)=Op0(G)'G=Op0p(G)isanabelianp0-groupwiththesamenumberofirreducibleBrauercharactersasG=Op0(G).Thus,jIBr(G=Op0(G))j=jIBr(G=Op0p(G))j=jG:Op0p(G)j=jG:Op0(G)jp0. WesaythatablockisbasicifitcontainsonlylinearBrauercharacters.ByTheorems 2.7 and 3.13 ,suchablockisisomorphictoF[G=Op0(G)]andisitsownbasicalgebra. 39

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CHAPTER4BASICALGEBRASOFBLOCKSOFP-SOLVABLEGROUPSWeareinterestedinthebasicalgebraofap-blockofanitep-solvablegroup.Letpbeaprimenumber.Throughout,letGbeanitep-solvablegroup,andusethenotationofChapter2. 4.1TheBasicAlgebraofaBlockofap-SolvableGroupThebasicalgebraofablockofanitep-solvablegrouphasaconcretedescriptionintermsofprimitiveidempotentsofap-complementandFongcharacters. Theorem4.1. LetGbeap-solvablegroupandB2Bl(G).LetH2Hallp0(G).Foreach'2IBr(B),let'2Irr(H)beaFongcharacterfor',andletd'=1 jHjXh2H'(1)')]TJ /F2 11.955 Tf 5.48 -9.68 Td[(h)]TJ /F9 7.97 Tf 6.59 0 Td[(1h2Z(FH),theprimitiveidempotentofFHassociatedto'.Letf'beaprimitiveidempotentinthealgebraFHd'.ThenthealgebraxFGxisabasicalgebraforFGeB,wherex=X'2IBr(B)f'. Proof. Foreach'2IBr(B),letM'betheirreducibleFG-modulewhichaffords',andletV'beaprojectivecoverofM'.ThenthebasicalgebraofBisE=EndFG0@M'2IBr(B)V'1Aop.(NotethatsincetheViareintheblockB,wehavethatEndFGeB0@M'2IBr(B)V'1Aop=EndFG0@M'2IBr(B)V'1Aop.)Now,byLemma 2.26 ,theprojectiveindecomposablecharacter'affordedbyV'isinducedfrom'.SincethealgebraFHissemisimple,FHd'isafullmatrixalgebrawithidentityd'.Sincef'isaprimitiveidempotentofFHd',theleftidealFHf'isanirreducibleFH-moduleaffordingcharacter'.Hence,V''IndFGFH(FHf')'FGFHFHf''FGf'. 40

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Notethateachf'livesinadifferentminimaltwo-sidedidealofFH,sothef'arepairwiseorthogonalidempotents.Therefore,E'EndFG0@M'2IBr(B)FGf'1Aop=EndFG(FGx)op'xFGxbyLemma 2.2 Denition4.2. LetB2Bl(G),whereGisanitegroup.Bisaprimepowerblockif'(1)isapowerofpforall'2IBr(B). Corollary4.3. LetGbeap-solvablegroupandB2Bl(G)beaprimepowerblock.LetH2Hallp0(G).Foreach'2IBr(B),let'2Irr(H)beaFongcharacterfor',andletd'=1 jHjXh2H'(1)')]TJ /F2 11.955 Tf 5.48 -9.68 Td[(h)]TJ /F9 7.97 Tf 6.59 0 Td[(1h2Z(FH),theprimitiveidempotentofFHassociatedto'.ThenthealgebraxFGxisabasicalgebraforFGeB,wherex=X'2IBr(B)d'. Proof. As'hasdegree'(1)p0=1,thealgebraFHd'isaone-dimensionalalgebra.Hence,d'isaprimitiveidempotentofFHd'.NowapplyTheorem 4.1 AnyidempotentconstructedasinTheorem 4.1 willbehenceforthreferredtoasabasicidempotentfortheblockalgebraFGeBorfortheblockB.NotethatifxisabasicidempotentforFGeB,thenxFGxisinfactasubalgebraofFGeB. 4.2Blocksofp-NilpotentGroupsIn[ 2 ],thebasicalgebraofanilpotentblockwithabeliandefectgroupisnotedtobeisomorphictothegroupalgebraoverthedefectgroupoftheblock.Inthissection,wepresentanelementary(atthelevelofgroupcharacters)proofofaspecialcaseofthisresult. 41

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Wewilldiscussthespecialcaseofablockofap-nilpotentgroupwithabelianquotient.LetGbeanitegroupwhichhasanormalHallp0-subgroupwhichcontainsthederivedsubgroupG0.Hence,G'HoP,wherePisanabelianp-groupandHisap0-group.Observethatinthiscase,thesetofp-regularelementsofGispreciselyH.Hence,asfunctions,'H='forall'2IBr(G). Lemma4.4. Thereisaone-to-onecorrespondencebetweentheG-conjugacyclassesofIrr(H)andIBr(G)givenby(f 1,..., tg)=tXi=1 i.Furthermore,eachblockB2Bl(G)containsauniqueirreducibleBrauercharacter'=tXi=1 i,andtheG-conjugatesofp-SylowsubgroupsofIG( 1)arethedefectgroupsofB. Proof. Letf = 1,..., tgIrr(H)beaG-conjugacyclass.LetI=IG( )betheinertiasubgroupof .ByFong'sTheorem[ 17 ,Theorem(10.20)],thereisauniqueblockbofIwhichcoverstheblockf gofHandIBr(b)=IBr(Ij ),andthedefectgroupsofbaretheSylowsubgroupsofI.Then,byGreen'sTheorem[ 17 ,Theorem(8.11)],IBr(b)=f g,where isnowviewedasanirreducibleBrauercharacterofI.Now,bytheFong-ReynoldsTheorem[ 17 ,Theorem(9.14)],thereisauniqueblock,namelyB=bGwhichcoverstheblockf gofH.Furthermore,everydefectgroupofbisadefectgroupofB,andBcontainsauniqueirreducibleBrauercharacter,namely'= G.ApplyingGreen'sTheoremagain,weconcludethat'=tXi=1 .Hence,iswell-dened.Now,if'2IBr(G),dene)]TJ /F1 11.955 Tf 6.32 0 Td[((')tobethesumoftheirreducibleconstituentsof'H.Thenitisclearthat)]TJ /F1 11.955 Tf 9.65 0 Td[(isaninverseto. WecannowdiscussthebasicalgebrasoftheblocksofG.Fix 2Irr(H),andletI=IG( ).Then,byLemma 4.4 determinesablockB2Bl(G)withIBr(B)='where'=tXi=1 iwheref = 1,..., tgistheG-conjugacyclassof inIrr(H).Now,since 42

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'(1)=t (1)=jG:Ij (1)andG=Iisap-groupandp(1),wehavethat isaFongcharacterfor'.Lete=1 jHjXh2H (1) (h)]TJ /F9 7.97 Tf 6.59 0 Td[(1)h2Z(FH),theprimitiveidempotentofZ(FH)correspondingto .ThenFHeisafullmatrixalgebra(ofdimension (1)2).ByTheorem 4.1 ,ifxisaprimitiveidempotentinthealgebraFHe,thenxFGxisabasicalgebrafortheblockB2Bl(G)whichcontains'.WewillshowthatwemaychooseaparticularprimitiveidempotentdinFHeinaveryspecialway.First,weneedagenerallinearalgebralemma. Lemma4.5. LetV6=0beanitedimensionalvectorspaceoverF.SupposeT1,...,Tr2EndF(V)arenilpotentlineartransformationswhichcommutepairwise.ThenthereexistsaprojectionS2EndF(V)withdimF(S(V))=1(inotherwords,Sisaprojectionontoaone-dimensionalsubspaceofV)andTiS=0forall1ir. Proof. Supposethisisfalse.Amongallcounterexamples,chooseVandT1,...,Tr2EndF(V)withrassmallaspossible.First,willweshowthatr>1.Indeed,ifr=0,thenwemaychooseStobeaprojectionontoanyone-dimensionalsubspaceofVsinceV6=0,andwearenotinacounterexample.Ifr=1,thenT1hasanontrivialkernel,andwemaychooseStobeprojectionontoanyone-dimensionalsubspaceofker(T1),andwearenotinacounterexample.Hence,r>1.Bytheminimalityofr,thereexistsaprojectionS02EndF(V)withdimF(S0(V))=1suchthatTiS0=0for1ir)]TJ /F1 11.955 Tf 11.95 0 Td[(1.Thus,06=S0(V)r)]TJ /F9 7.97 Tf 6.58 0 Td[(1\i=1ker(Ti).Inparticular,K=r)]TJ /F9 7.97 Tf 6.58 0 Td[(1\i=1ker(Ti)6=0.SinceTrcommuteswithT1,...,Tr)]TJ /F9 7.97 Tf 6.59 0 Td[(1,wehavethatTr(ker(Ti))ker(Ti) 43

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forall1ir)]TJ /F1 11.955 Tf 12.24 0 Td[(1.Inparticular,Tr(K)K.SinceTrisnilpotent,ker(TrjK)6=0,sinceotherwise,everypowerofTrjKisanisomorphismofKandthiswouldyieldK=0,whichisacontradiction.Therefore,K\ker(Tr)6=0.Finally,06=ker(Tr)\K=r\i=1ker(Ti).Thus,choosingS1aprojectionontoaone-dimensionalsubspaceofr\i=1ker(Ti)givesTiS1=0forall1ir,whichisacontradiction,sincethechoiceofV,T1,...,Trisacounterexampletothestatementofthelemma. Ifg2I,theneg=e.SinceconjugationbygisagroupautomorphismofH,themapg:FHe!FHegivenbyg(x)=xgforallx2FHeisanF-algebraautomorphism.BytheNoether-SkolemTheorem[ 10 ,Theorem6.7],gisaninnerautomorphismofFHe.Hence,thereexistsanelementg2(FHe)suchthatg(x)=g)]TJ /F9 7.97 Tf 6.59 0 Td[(1xgforallx2FHe.Thenextlemmagivesanadditionalconditionongsothatitisuniquelydeterminedbyg.LetP0=CP(H)andP1=IP( ).ThenP0P1I.ByLemma 4.4 ,P1isadefectgroupofB.WeshouldshowthatthebasicalgebraofBisisomorphictoFP1. Lemma4.6. Writepk=[P1:P0].Forallg2P1,thereexistsauniqueeg2(FHe)suchthategpk=eandxg=xegforallx2FHe.Furthermore,themapP1!(FHe)givenbyg7!egisagrouphomomorphism.Inparticular,egeh=ehegandeg)]TJ /F9 7.97 Tf 6.58 0 Td[(1=gg)]TJ /F9 7.97 Tf 6.58 0 Td[(1forallg,h2P1. Proof. Wewillrstproveexistence.Letg2(FHe)besuchthatxg=xgforallx2FHe.Then,asgpk2P0,wehavethatxgpk=xgpk=xforallx2FHe.Thus,sinceFHeisafullmatrixalgebra,gpk2Z(FHe),andsogpk=eforsome2F.Leteg=gwhere2Fissuchthatpk=)]TJ /F9 7.97 Tf 6.58 0 Td[(1.Thenegpk=e. 44

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Nowweproveuniqueness.Supposeeg1,eg22(FHe)andxeg1=xeg2forallx2FHe.Thenxeg1eg2)]TJ /F12 5.978 Tf 5.75 0 Td[(1=xforallx2FHe.Therefore,eg1eg2)]TJ /F9 7.97 Tf 6.58 0 Td[(12Z(FHe),sothateg1=eg2,some2F.Ifeg1pk=e=eg2pk,thene=eg1pk=(eg2)pk=pke.Thus,isapk-throotofunityinF.SinceFhascharacteristicp,=1,andthuseg1=eg2.Wewillnowshowthateh)]TJ /F9 7.97 Tf 6.59 0 Td[(1=gh)]TJ /F9 7.97 Tf 6.58 0 Td[(1forallh2P1.Itisclearthatxeh)]TJ /F12 5.978 Tf 5.76 0 Td[(1=xh)]TJ /F12 5.978 Tf 5.76 0 Td[(1=xgh)]TJ /F12 5.978 Tf 5.75 0 Td[(1forallx2FHe.Now,eh)]TJ /F9 7.97 Tf 6.59 0 Td[(1pk=eh)]TJ /F9 7.97 Tf 6.59 0 Td[(1pke=eh)]TJ /F9 7.97 Tf 6.58 0 Td[(1pkehpk=eh)]TJ /F9 7.97 Tf 6.59 0 Td[(1ehpk=epk=e.Byuniquenessofgh)]TJ /F9 7.97 Tf 6.59 0 Td[(1,thisimpliesthateh)]TJ /F9 7.97 Tf 6.58 0 Td[(1=gh)]TJ /F9 7.97 Tf 6.59 0 Td[(1.Wewillnowshowthategeh=ehegforallg,h2P1.Indeed,forallx2FHe,xeg=xg=xh)]TJ /F12 5.978 Tf 5.76 0 Td[(1gh=xgh)]TJ /F12 5.978 Tf 5.76 0 Td[(1egeh=xeh)]TJ /F12 5.978 Tf 5.75 0 Td[(1egeh.Furthermore,(eh)]TJ /F9 7.97 Tf 6.59 0 Td[(1egeh)pk=eh)]TJ /F9 7.97 Tf 6.59 0 Td[(1egpkeh=eh)]TJ /F9 7.97 Tf 6.58 0 Td[(1eKeh=e=egpk.Hence,byuniqueness,eh)]TJ /F9 7.97 Tf 6.58 0 Td[(1egeh=eg.Finally,wewillshowthatthemapg7!egisagrouphomomorphismP1!(FHe).Letg,h2P1.Wewillshowthategeh=fgh.Bytherstpartoftheproof(existenceanduniqueness),itsufcestoshowthatxegeh=xghforallx2FHeand(egeh)pk=e.Ifx2P1,wehavexegeh=(xeg)eh=(xg)h=xgh.Also,(egeh)pk=egpkehpk=ee=esinceegandehcommute. 45

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Therefore,byLemma 4.6 ,foreachg2P1,wemaydeneegtobetheuniqueelementof(FHe)suchthategpk=e(wherepk=jP1:P0jandxg=xegforallx2FHe. Lemma4.7. Thereexistsaprimitiveidempotentd2FHesuchthategd=dforallg2P1. Proof. Letpk=[P1:P0].LetC=feg)]TJ /F2 11.955 Tf 12.08 0 Td[(ejg2P1gFHe.ByLemma 4.6 ,theelementsofCcommutepairwise.SinceFhascharacteristicpande2Z(FHe),wehave(eg)]TJ /F2 11.955 Tf 11.96 0 Td[(e)pk=egpk)]TJ /F2 11.955 Tf 11.96 0 Td[(epk=e)]TJ /F2 11.955 Tf 11.96 0 Td[(e=0,soeveryelementofCisnilpotent.Hence,byLemma 4.5 ,thereexistsaprimitiveidempotentd2FHesuchthat(eg)]TJ /F2 11.955 Tf 11.96 0 Td[(e)d=0forallg2P1.Hence,0=(eg)]TJ /F2 11.955 Tf 11.96 0 Td[(e)d=egd)]TJ /F2 11.955 Tf 11.96 0 Td[(ed=egd)]TJ /F2 11.955 Tf 11.95 0 Td[(dforallg2P1. Lemma4.8. Forallx2FP1,dxd=xd.Inparticular,dFP1d=FP1d,andFP1disanalgebrawithidentityd. Proof. Itsufcestoshowthatdgd=gdforallg2P1.Ifg2P1,thendgd=gg)]TJ /F9 7.97 Tf 6.59 0 Td[(1dgd=geg)]TJ /F9 7.97 Tf 6.58 0 Td[(1degd=gdd=gdbyLemma 4.7 Lemma4.9. WehaveFP1d'FP1asF-algebras. Proof. Considerthemap:FP1!FP1d 46

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givenby(x)=xdforallx2FP1.Thenitisclearthatisasurjectivehomomorphismofvectorspaces.Hence,dimF(FP1d)dimF(FP1)=jP1j.Then,thesetfgdjg2P1gFP1disacollectionofjP1jlinearlyindependentelementsofFP1dsinceg1dandg2dinvolvedifferentelementsofGforg16=g22P1.Hence,dimF(FP1d)jP1jandsodimF(FP1d)=jP1j.Therefore,isanisomorphismofvectorspaces.Wewillshowthatisanalgebrahomomorphism.Indeed,ifx,y2FP1,thenbyLemma 4.8 ,(x)(y)=xdyd=xyd=(xy).Since,(1)=d,whichistheidentityofFP1d,isanalgebrahomomorphism. Lemma4.10. Ifx2GnI,thendxd=0. Proof. First,observethatifx2GnI,thenxex)]TJ /F9 7.97 Tf 6.59 0 Td[(1isaprimitiveidempotentofZ(FH)whichisdifferentfrome.Hence,exe=exe(x)]TJ /F9 7.97 Tf 6.59 0 Td[(1x)=e(xex)]TJ /F9 7.97 Tf 6.59 0 Td[(1)x=0x=0.Sowehavedxd=dexed=0. Lemma4.11. Ify2H,thendydisanF-multipleofd. Proof. SincedisanidempotentinFH,wehave,byLemma 2.3 ,thatdFHd'(EndFH(FHd))op. 47

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AsFHdisasimpleFH-moduleandFisalgebraicallyclosed,EndFH(FHf)'FbySchur'sLemma[ 17 ,Lemma(1.4)].Thus,dFHd'Fop'F.Inparticular,dimF(dFHd)=1.Sinced6=0,dFHd=spanF(d),andwearedone. Lemma4.12. WehavedFGd=FP1d. Proof. LetX=fdxydjx2H,y2Pg.SinceG=HoP,dFGd=spanF(X).WewillshowthatspanF(X)=FP1d.Letx2H.Lety2PnP1.Then,sincexy=2I,dxyd=0byLemma 4.10 .Ify2P1,thendxyd=dxdyd=ydforsome2FbyLemmas 4.8 and 4.11 .Hence,spanF(X)=FP1d. Wehaveprovedthenalresultofthissection. Theorem4.13. LetB2Bl(G),whereG=HoP,wherep-jHjandPisap-group.ThenBcontainsauniqueirreducibleBrauercharacter'.Let beanirreducibleconstituentof'H.ThenBhasdefectgroupD=IP( ).IfDisabelian,thenthebasicalgebraofBisFD.Furthermore,theblockalgebraFGeB'Mat( (1),FD)asF-algebras. 4.3BasicIdempotentsforthePrincipalBlockWithTwoIrreducibleBrauerCharactersofap-SolvableGroupNow,wewillrestrictourattentiontotheprincipalblockBofGandrenetheresultsofSection4.1inthecasethatBcontainsexactlytwoirreducibleBrauercharacters.Westartbystating[ 18 ,Theorem3.1],whichclassiesp-solvablegroupswhoseprincipalblockcontainsexactlytwoirreducibleBrauercharacters. 48

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Theorem4.14(NinomayaandWada). LetGbeap-solvablegroupandB0theprincipalblockofFG.Thenthefollowingareequivalent. ThenumberofirreducibleBrauercharactersinB0is2. G=Op0pp0p(G)and 1. ifGhasp-length1,thenpisoddandjG:Op0p(G)j=2,and 2. ifGhasp-length2,thenoneofthefollowingholds: (a) p=2,G=O202(G)'E32oZ8. (b) p=2,G=O202(G)'E32oQ8,whereQ8isthequaterniongroupoforder8. (c) p=2,G=O202(G)'E32oS16,whereS16isthesemidihedralgroupoforder16. (d) p=2,G=O202(G)'ZqoZ2n,whereq=2n+1isaFermatprime. (e) p=2n)]TJ /F1 11.955 Tf 11.95 0 Td[(1(aMersenneprime),G=O202(G)'E2noZp.Itisalsoworthstating[ 11 ,TheoremA],whichimpliesthatinasolvablegroup,ifablockcontainsexactlytwoirreducibleBrauercharacters,thenthep0-partsofthedegreesofthesetwocharactersareequal. Theorem4.15(Isaacs). LetGbesolvableandsupposethatBisap-blockofG.AssumethatIBr(B)=f,gandthat(1)>(1).TheneitherpisaMersenneprimeand(1)=(1)=p,orelsep=2and(1)=(1)isapowerof2thatisequalto8,orelseisoftheformq)]TJ /F1 11.955 Tf 11.95 0 Td[(1,whereqisaFermatprime.Inallcases,jGjiseven.Westartwithalemmawhichwillallowustodealwiththep-length1case. Lemma4.16. SupposeGisap-solvablegroupwithprincipalblockBandthatG=Op0p(G)isanabeliangroup.ThenBisbasic. Proof. Let'2IBr(B).By[ 17 ,Theorem(10.20)],Op0(G)ker(').Hence,by[ 17 ,Lemma(2.32)],Op(G=Op0(G))=Op0p(G)=Op0(G)iscontainedinthekernelof'viewedasaBrauercharacterofG=Op0(G).ItfollowsthatOp0p(G)ker(').AsG=Op0p(G)isabelian,alloftheBrauercharactersinBarelinear.Hence,byLemma 2.7 ,Bisbasic. 49

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Corollary4.17. SupposeGisap-solvablegroupandthatBistheprincipalblockofG.IfGhasp-length1andBcontainsexactlytwoBrauercharactersofG,thenB'F[G=Op0(G)]andBisbasic. Proof. By 3.13 ,B'F[G=Op0(G)].Then,by 4.14 ,G=Op0p(G)'Z2anditfollowsfromLemma 4.16 thatBisbasic. Thep-length1caseisnowcomplete.Wewillnowdiscussthep-length2case. Lemma4.18. SupposeK'QoPwhereQisanabelianp0-groupandPisap-groupactingtransitivelyonQ.ThenKhasp-BrauercharactertableK0=Q 1Q 1Q 11' jQj)]TJ /F1 11.955 Tf 17.94 0 Td[(1)]TJ /F1 11.955 Tf 9.3 0 Td[(1.Furthermore,'=X2Irr(Q)nf1Qg. Proof. ItisclearthatK0=QandthatQisaK-conjugacyclass.Also,sincePactstransitivelyonQandQisanabeliangroup,wehavethattheactionofPonIrr(Q)istransitiveonthenontrivialcharactersinIrr(Q).Let'2IBr(K),'6=1K0.Let2Irr(Q)nf1Qg.ByGreen'sTheorem[ 17 ,Theorem(8.11)],thereexistsaunique'2IBr(K)lyingover.Furthermore,'QisthesumofthedistinctK-conjugatesof.AsanynontrivialBrauercharacterliesover,'istheuniquenontrivialBrauercharacterofK.Furthermore,as'='Q,wehavethat'isthesumofthenontrivialirreduciblecharactersofQ.Let16=x2Q.Wewillcompute'(x).BytheSecondOrthogonalityRelation[ 12 ,Theorem(2.18)],'(x)=X2Irr(Q)nf1Qg(x)=X2Irr(Q)(x))]TJ /F1 11.955 Tf 11.95 0 Td[(1Q(1)=0)]TJ /F1 11.955 Tf 11.96 0 Td[(1=)]TJ /F1 11.955 Tf 9.29 0 Td[(1.AsQisabelianandtherearejQj)]TJ /F1 11.955 Tf 18.28 0 Td[(1irreducibleconstituentsof'Q,wehavethat'hasdegreejQj)]TJ /F1 11.955 Tf 17.93 0 Td[(1. 50

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Theorem4.19. SupposeGisanitep-solvablegroupwithOp0(G)=1andG=Op(G)'QoPwhereQisanabelianp0-groupoforderatleast3andPisap-groupactingtransitivelyonQ.Then ifH2Hallp0(G),thenH'Q. forallH2Hallp0(G)andall1H6=2Irr(H),thealgebraxFGxisabasicalgebraforFG,wherex=Xh2H(1+(h))h2FG. Proof. Let:G!QoPbethenaturalprojection.LetH2Hallp0(G).ItisclearthatjH:H!Qisanisomorphism.AsOp(G)ker(')[ 17 ,Lemma(2.32)]forall'2IBr(G),theirreducibleBrauercharactersofGarepreciselyfj2IBr(G=Op(G))g.ByLemma 4.18 ,GhasexactlytwoirreducibleBrauercharacters,thetrivialcharacter1G0,andanontrivialBrauercharacter.SinceOp0(G)istrivial,Ghasauniqueblockby[ 17 ,Theorem(10.20)].Hence,byTheorem 4.3 (andthefactthatjQj)]TJ /F1 11.955 Tf 18.01 0 Td[(1mustbeapowerofp),xFGxisabasicalgebraforFG,wherex2FGisthesumoftheFongidempotentsfor1G0and.Clearly,1HisaFongcharacterfor1G0.ByLemma 4.18 ,anynontrivialirreduciblecharacterofHisaFongcharacterfor'.Fixanontrivial2Irr(H).Thenx=e1H+e=1 jHjXh2H1H(1)1H(h)h+1 jHjXh2H(1)(h)h=1 jHjXh2H(1+(h))h.Now,sincejQj3andPactstransitivelyonQ,wehavethat16=jQjisapowerofp.Therefore,jHj=jQjiscongruentto1modulop.ItfollowsthatjHj=1,andweobtainthedesiredresultbythepreviousline. Corollary4.20. SupposeGisanitep-solvablegroupwithG=Op0p(G)'QoPwhereQisanabelianp0-groupoforderatleast3andPisap-groupactingtransitivelyonQ.LetBbetheprincipalblockofG.Then 51

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ifH2Hallp0(G=Op0(G)),thenH'Q. forallH2Hallp0(G=Op0(G))andall1H6=2Irr(H),thealgebraxF[G=Op0(G)]xisabasicalgebraforB,wherex=Xh2H(1+(h))h2F[G=Op0(G)]. Proof. By[ 20 ],BisisomorphictoF[G=Op0(G)].Hence,wewillshowthatthegroupG=Op0(G)satisesthehypothesesofTheorem 4.19 .AsOp0(G=Op0(G))istrivial,wehavethatOp0p(G=Op0(G))=Op(G=Op0(G)).Hence,(G=Op0(G))=Op(G=Op0(G))=(G=Op0(G))=(Op0p(G)=Op0(G))'G=Op0p(G)'QoP.TheresultisobtainedbyapplyingTheorem 4.19 toG=Op0(G). Corollary4.21. SupposeGisap-solvablegroupandBistheprincipalblockofG.IfGhasp-length2andBcontainsexactlytwoirreducibleBrauercharacters,then ifH2Hallp0(G=Op0(G)),thenHisanelementaryabeliangroup. forallH2Hallp0(G=Op0(G))andall1H6=2Irr(H),thealgebraxF[G=Op0(G)]xisabasicalgebraforB,wherex=1 jHjXh2H(1+(h))h2F[G=Op0(G)]. Proof. By[ 18 ,Theorem3.1],G=Op0p(G)'QoPwhereQisanelementaryabelianp0-groupoforderatleast3andPisap-group.SinceOp0p(G)ker(')forall'2B(by[ 17 ,Lemma(10.20)]and[ 17 ,Lemma(2.32)]),theirreducibleBrauercharactersinBmaybeviewedasirreducibleBrauercharactersofG=Op0p(G).ByGreen'sTheorem[ 17 ,Theorem(8.11)],PmustacttransitivelyonthenontrivialirreduciblecharactersofQ,andhencePactstransitivelyonQ.ThedesiredresultisobtainedbyapplyingCorollary 4.20 52

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Finally,wesummarizeourcalculationofthebasicalgebraintheeventthattheprincipalblockcontainsexactlytwoirreducibleBrauercharacters. Theorem4.22. SupposeGisap-solvablegroup,BistheprincipalblockofG,andthatBcontainsexactlytwoirreducibleBrauercharacters.Then B'F[G=Op0(G)]. IfGhasp-length1,thenBisbasic. IfGhasp-length2,thenBhasbasicalgebraxF[G=Op0(G)]x,wherex=Xh2H(1+(h))h2F[G=Op0(G)]foranyH2Hallp0(G=Op0(G))andany1H6=2Irr(H).Necessarily,Hiselementaryabelian. Corollary4.23. SupposeGisap-solvablegroupandBisablockofGcontainingexactlytwoirreducibleBrauercharacters,atleastoneofwhichislinear.Then BhasbasicalgebraF[G=Op0(G)]ifbothBrauercharactersinBarelinear. IfBcontainsanon-linearBrauercharacter,BhasbasicalgebraxF[G=Op0(G)]x,wherex=Xh2H(1+(h))h2F[G=Op0(G)]foranyH2Hallp0(G=Op0(G))andany1H6=2Irr(H). Proof. FollowsimmediatelyfromTheorems 4.22 and 3.13 53

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CHAPTER5IMPLEMENTATIONINGAPInthischapter,wediscusstheimplementationofTheorem 4.1 inGAP(see[ 8 ]). 5.1TheAlgorithmIntheory,thealgorithmforcomputingabasicidempotentx2FGofablockBofanitep-solvablegroupGisjusttheproofofTheorem 4.1 .Weoutlinethegeneralprocesshereandthendiscussthetechnicaldetailsoftheimplementation. Algorithm5.1. INPUT:Aprimep,anitep-solvablegroupG,analgebraicallyclosedeldFofcharacteristicp,andap-blockB2Bl(G).OUTPUT:AnidempotentxinFGsuchthatxFGxisabasicalgebraofB. 1. ChooseaHallp0-subgroupHofG. 2. Foreach'2IBr(B),consider'H.Chooseanirreducibleconstituent'of'Hwith'(1)='(1)p0. 3. Computetheprimitiveidempotentd'=1 jHjXh2H'(h)]TJ /F9 7.97 Tf 6.59 0 Td[(1)hofZ(FH)correspondingto'. 4. Findaprimitiveidempotentf'inthefullmatrixalgebraFHd' 5. Setx=X'2IBr(B)d'.OnedifcultyofimplementingtheabovealgorithminGAPisthenatureoftheeldF.AsonecannotimplementalgebraicclosuresofeldsinGAP,aniteeldmustsufce.WesaythataniteeldFislargeenoughforGifFcontainsallofthejGjp0=jHj-rootsofunity.AsthevaluesoftheirreducibleBrauercharactersofGaresumsofjHj-throotsofunity,suchaeldwillsufceforourcalculations.Hence,wetakeFtobetheeldGF(pn)wherenisthesmallestintegersuchthatjHjdividespn)]TJ /F1 11.955 Tf 11.96 0 Td[(1.IntheGAPimplementationoftheabovealgorithm,wemustalsocomputed'asinStep3.ThealgebraicintegerjHj)]TJ /F9 7.97 Tf 6.59 0 Td[(1hasanaturalembedding(jHj)]TJ /F9 7.97 Tf 6.58 0 Td[(1modp)in 54

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theprimeeldofF),but'(h)]TJ /F9 7.97 Tf 6.59 0 Td[(1)doesnotnecessarilyhavesuchanembedding.However,thisisnotamajorproblem.Noticethat'jhh)]TJ /F12 5.978 Tf 5.76 0 Td[(1iisapositiveintegerlinearcombinationoflinearcharacterssXi=1aii,ai>0,i2Irr(h)]TJ /F9 7.97 Tf 6.58 0 Td[(1)isalinearcharacter.Sinceiisalinearcharacter,wehavethati(h)]TJ /F9 7.97 Tf 6.59 0 Td[(1)isajHj-throotofunity,andishenceequaltosomepowerofE(jHj),whereE(m)=e2i m.ExtendingthecorrespondenceE(jHj)7!Z(pn)pn)]TJ /F12 5.978 Tf 5.76 0 Td[(1 jHj(whereZ(pn)isGAP'spreferredgeneratorforF)topowersofE(jHj)givesanembeddingofi(h)]TJ /F9 7.97 Tf 6.59 0 Td[(1)inFinthefollowingway:ifi(h)]TJ /F9 7.97 Tf 6.58 0 Td[(1)=E(jHj)w2C,seti(h)]TJ /F9 7.97 Tf 6.58 0 Td[(1)=Z(pn)pn)]TJ /F21 5.978 Tf 5.76 0 Td[(1 jHjw2F.Thus,')]TJ /F2 11.955 Tf 5.48 -9.68 Td[(h)]TJ /F9 7.97 Tf 6.59 0 Td[(1=sXi=1aii)]TJ /F2 11.955 Tf 5.48 -9.68 Td[(h)]TJ /F9 7.97 Tf 6.59 0 Td[(1=sXi=1(aimodp)i)]TJ /F2 11.955 Tf 5.48 -9.68 Td[(h)]TJ /F9 7.97 Tf 6.59 0 Td[(1.Oncewehavecomputedd',theproblemremainsofndingaprimitiveidempotentf'2FHd'.In[ 4 ],theauthorsdiscussamethodforcomputingprimitiveidempotentsinmatrixalgebras,butthismethodiscurrentlynotimplemented.Forsmallexamples,thefollowingbruteforcealgorithmiseffectiveforndingaprimitiveidempotentf'.NotethatbyWedderburn'sTheorem,FHd''EndF(M'),whereMistheirreducibleFH-modulewhosecharacteris'.Equivalently,M'isanirreducibleFH-submodule(orirreducibleleftideal)ofFHd'. Algorithm5.2. INPUT:ThefullmatrixalgebraFHd',theFongcharacter'.OUTPUT:Aprimitiveidempotentf'2FHd'. 1. LetxbearandomelementofFHd'. 2. IfdimF(FHd'x)='(1)2)]TJ /F10 11.955 Tf 13.13 0 Td[('(1),thenM=FHd'=FHd'xisanirreducibleFHd'-module.Otherwise,gobacktoStep1. 3. LetXMbeabasis,andletZFHd'beageneratingset.Letbethealgebraisomorphism:FHd'!EndF(M)givenby(z)isthematrixwithrespecttothebasisXfortheactionofzonMgivenbyleftmultiplication. 55

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4. LetPbethe'(1)'(1)matrixoverFwith1intheupperleft-handcornerand0everywhereelse.ThenPisaprimitiveidempotentofEndF(I). 5. Setf=)]TJ /F9 7.97 Tf 6.58 0 Td[(1(P).AnotherissuewithimplementingAlgorithm 5.1 inGAPisndingap-complementH.InGAPversion4.5.6,functionalityforcomputingap-complementexistsviatheHallSubgrouporSylowComplementcommands.However,inthecurrentstateofGAP,thesecommandsareonlyfunctionalforgroupswhicharesolvable.Hence,intheGAPimplementation,wecanonlycalculatethebasicalgebraforgroupswhicharesolvable,andnotthelargerclassofp-solvablegroups. 5.2SourceCodeThissectioncontainsthesourcecodefortheGAPimplementationofAlgorithm 5.1 .ThecodemaybeinputintoGAPusingtheReadfunction.ThemainfunctionisBasicAlgebra(G,B,p)whichtakesasinputanitesolvablegroupG,aprimep,andap-blockBgivenasanelementinthelistBlocksInfo(CharacterTable(G,p)).ThefunctionBasicAlgebra(G,B,p)outputsaGAPrecordcontainingthefollowingcomponents: groupisthegroupG, pcompisap-complementsubgroupH, primeistheprimep, blockisthep-blockB, fieldisaeldFofcharacteristicpwhichislargeenoughforG, groupalgebraisthegroupalgebraFG, basidemisanidempotentx2FGsuchthatxFGxisabasicalgebraofB,and basicalgebraisthebasicalgebraxFGxofB.Hereisthesourcecode. 56

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#GAPSOURCECODEFORBASICALGEBRAPROGRAMModPReduction:=function(x,sizeF,sizeH)#computesthemod-preductioninFofx,whichisa|H|-throotof1localp,n,zeta,beta;p:=FactorsInt(sizeF)[1];n:=Log(sizeF,p);zeta:=Z(p^n)^((p^n-1)/sizeH);beta:=0;whilebeta>=0doifx=E(sizeH)^betathenbreak;fi;ifx<>E(sizeH)^betathenbeta:=beta+1;fi;od;returnzeta^beta;end;pPrimePart:=function(n,p)#computesthep'-partoftheintegernlocalfactors,ppart,x;factors:=FactorsInt(n);ifnotpinfactorsthenreturnn;fi; 57

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ppart:=1;forxinfactorsdoifx=pthenppart:=ppart*x;fi;od;returnn/ppart;end;FongIdemMod:=function(FG,G,H,phi,p)#computesaFongidempotentforaBrauercharacterphiofGlocalalpha,constituents,theta,f_alpha,FH,FHf,basisM,pi,M,X,J,count,dimensiondeccheck,rightdimensioncheck,FHf_alpha,x,j,fongdeg,sizeF,k,a,s,matak,lambda,imagebasisI,coeffsakxj,h,f,F,basisI,primidemmat,Phi,MatAlg,proj,imgproj,gensFHf,z,imgPhi,i;fongdeg:=pPrimePart(DegreeOfCharacter(phi),p);constituents:=ConstituentsOfCharacter(RestrictedClassFunction(phi,H));forthetainconstituentsdoifDegreeOfCharacter(theta)=fongdegthenalpha:=Irr(H)[Position(Irr(H),theta)];break;fi;od;sizeF:=RootInt(Size(FG),Dimension(FG));f_alpha:=(One(FG)/Size(H)*One(FG))*Sum(Elements(H),h->DegreeOfCharacter(alpha)*One(FG)*Sum(ConstituentsOfCharacter(RestrictedClassFunction(alpha,Subgroup(H,[h]))),lambda->ScalarProduct(CharacterTable(Subgroup(H,[h])), 58

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lambda,RestrictedClassFunction(alpha,Subgroup(H,[h])))*One(FG)*ModPReduction(h^lambda,sizeF,Order(H)))*(h^-1)^Embedding(G,FG));FH:=Subalgebra(FG,Image(Embedding(G,FG),Elements(H)));FHf:=FH*f_alpha;ifDimension(FHf)=1thenreturnf_alpha;fi;F:=UnderlyingField(FG);count:=0;rightdimensioncheck:=0;dimensiondeccheck:=0;forxinFHfdo#findamaximalsubmoduleofregularmoduleJ:=LeftIdeal(FHf,[x]);count:=count+1;ifDimension(J)=(fongdeg^2-fongdeg)thendimensiondeccheck:=dimensiondeccheck+1;break;fi;od;rightdimensioncheck:=rightdimensioncheck+1;MatAlg:=FullMatrixAlgebra(F,fongdeg);pi:=NaturalHomomorphismBySubspace(FHf,J);M:=Image(pi);gensFHf:=GeneratorsOfAlgebra(FHf);basisM:=Basis(M);imgPhi:=[];forkin[1..Length(gensFHf)]do 59

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matak:=IdentityMat(fongdeg,F);forjin[1..Length(basisM)]docoeffsakxj:=Coefficients(basisM,(gensFHf[k]*PreImagesRepresentative(pi,basisM[j]))^pi);foriin[1..Length(basisM)]domatak[i][j]:=coeffsakxj[i];od;od;imgPhi[k]:=matak;od;Phi:=AlgebraHomomorphismByImages(FHf,MatAlg,gensFHf,imgPhi);proj:=IdentityMat(fongdeg,F);forsin[2..fongdeg]doproj[s][s]:=Zero(F);od;returnPreImage(Phi,proj);end;BasIdem:=function(FG,G,H,B,p)#computesabasicidempotentofBasasumofFongidempotentslocale,i;e:=Sum(B.modchars,i->FongIdemMod(FG,G,H,Irr(G,p)[i],p));returne;end;PComplement:=function(G,p)#computesaHallp'-subgroupofG 60

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localpi,hallsub,K,phi;K:=SmallGroup(IdSmallGroup(G));pi:=Set(FactorsInt(Order(G)));ifnotpinpithenreturnG;fi;ifSize(pi)=1thenreturnTrivialSubgroup(G);fi;Remove(pi,Position(pi,p));phi:=IsomorphismGroups(K,G);hallsub:=HallSubgroup(K,pi);hallsub:=Image(phi,hallsub);returnhallsub;end;BasicAlgebra:=function(G,B,p)#mainfunctionwhichcomputesthebasicalgebraofBlocalF,FG,H,n,bas,basidem;H:=PComplement(G,p);n:=1;whilen>=1doif(p^n-1)/Size(H)inIntegersthenbreak;fi;ifnot(p^n-1)/Size(H)inIntegersthenn:=n+1; 61

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fi;od;F:=GF(p^n);FG:=GroupRing(F,G);basidem:=BasIdem(FG,G,H,B,p);bas:=rec(group:=G,block:=B,prime:=p,field:=F,groupalgebra:=FG,idempotent:=basidem,basicalgebra:=basidem*FG*basidem,pcomp:=H);returnbas;end; 5.3SampleOutputInthissection,weusetheGAPcodegiveninSection5.2toproducealistofblocksBofsolvablegroupsGsatisfyingthefollowing. jGj<96, Gisnotp-nilpotentorBhasnonabeliandefectgroups, Bisnotbasic,and ifBcontainsalinearBrauercharacter,thenBistheprincipalblockofG.Table 5-1 givestheGAPSmallGroupID(givenbyIdSmallGroup(G)),theprimep,theblockIDofB(whichisthepositionofBinBlocksInfo(CharacterTable(G,p)),andthedimensionofBanditsbasicalgebrabas(B). 62

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Table5-1. DimensionsofBlocksandTheirBasicAlgebrasinSmallSolvableGroups SmallGroupIDpBlockIDBrauerdegreesdimF(B)dimF(bas(B)) [24,4]33[2]123[24,6]33[2]123[24,8]33[2]123[24,12]21[1,2]2411[24,12]32[3]91[24,12]33[3]91[30,3]32[2]123[30,3]33[2]123[30,3]52[2]205[40,4]53[2]205[40,6]53[2]205[40,8]53[2]205[42,5]32[2]123[42,5]33[2]123[42,5]34[2]123[42,5]72[2]287[48,5]35[2]123[48,5]36[2]123[48,6]33[2]123[48,6]34[2]123[48,6]35[2]123[48,7]33[2]123[48,7]34[2]123[48,7]35[2]123[48,8]33[2]123[48,8]34[2]123[48,8]35[2]123[48,10]35[2]123[48,10]36[2]123[48,12]35[2]123[48,12]36[2]123[48,13]35[2]123[48,13]36[2]123[48,14]35[2]123[48,14]36[2]123[48,15]22[2]328[48,15]33[2]123[48,15]34[2,2]246[48,16]22[2]328[48,16]33[2]123[48,16]34[2,2]246[48,17]22[2]328[48,17]33[2]123[48,17]34[2,2]246 63

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Table 5-1 .Continued SmallGroupIDpBlockIDBrauerdegreesdimF(B)dimF(bas(B)) [48,18]22[2]328[48,18]33[2]123[48,18]34[2,2]246[48,19]35[2]123[48,19]36[2]123[48,28]21[1,2]4822[48,28]32[2,2]246[48,28]33[3]91[48,28]34[3]91[48,29]21[1,2]4822[48,29]32[2,2]246[48,29]33[3]91[48,29]34[3]91[48,30]21[1,2]4822[48,30]33[3]91[48,30]34[3]91[48,30]35[3]91[48,30]36[3]91[48,34]35[2]123[48,34]36[2]123[48,36]35[2]123[48,36]36[2]123[48,37]35[2]123[48,37]36[2]123[48,38]22[2]328[48,38]35[2,2]246[48,39]22[2]328[48,39]35[2,2]246[48,40]22[2]328[48,40]35[2,2]246[48,41]22[2]328[48,41]35[2,2]246[48,43]35[2]123[48,43]36[2]123[48,48]21[1,2]4822[48,48]33[3]91[48,48]34[3]91[48,48]35[3]91[48,48]36[3]91[56,3]73[2]287[56,5]73[2]287[56,7]73[2]287[60,3]33[2]123[60,3]34[2]123 64

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Table 5-1 .Continued SmallGroupIDpBlockIDBrauerdegreesdimF(B)dimF(bas(B)) [60,3]35[2]123[60,3]36[2]123[60,3]53[2]205[60,3]54[2]205[60,7]33[4]483[60,7]52[2,2]4010[60,8]33[2,2]246[60,8]34[2,2]246[60,8]53[2,2]4010[60,12]33[2]123[60,12]34[2]123[60,12]35[2]123[60,12]36[2]123[60,12]53[2]205[60,12]54[2]205[66,3]32[2]123[66,3]33[2]123[66,3]34[2]123[66,3]35[2]123[66,3]36[2]123[66,3]112[2]4411[70,3]52[2]205[70,3]53[2]205[70,3]54[2]205[70,3]72[2]287[70,3]73[2]287[72,4]33[2]369[72,6]33[2]369[72,8]33[2]369[72,15]21[1,2]2411[72,15]22[2,2,2]4812[72,15]32[3,3]546[72,22]32[2]369[72,23]32[2]369[72,24]32[2]369[72,26]33[2]369[72,28]33[2]369[72,30]33[2]369[72,31]33[2]369[72,33]33[2]369[72,35]33[2]369[72,40]31[1,1,1,1,2]7241[72,41]31[1,1,1,1,2]7241[72,42]21[1,2]2411 65

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Table 5-1 .Continued SmallGroupIDpBlockIDBrauerdegreesdimF(B)dimF(bas(B)) [72,42]32[3]273[72,42]33[3]273[72,43]21[1,2]2411[72,43]22[2,2,2]4812[72,43]32[3,3]546[72,44]22[2,2,2]4812[72,44]32[3,3]546[78,5]32[2]123[78,5]33[2]123[78,5]34[2]123[78,5]35[2]123[78,5]36[2]123[78,5]37[2]123[78,5]132[2]5213[80,5]55[2]205[80,5]56[2]205[80,6]53[2]205[80,6]54[2]205[80,6]55[2]205[80,7]53[2]205[80,7]54[2]205[80,7]55[2]205[80,8]53[2]205[80,8]54[2]205[80,8]55[2]205[80,10]55[2]205[80,10]56[2]205[80,12]55[2]205[80,12]56[2]205[80,13]55[2]205[80,13]56[2]205[80,14]55[2]205[80,14]56[2]205[80,15]22[2]328[80,15]23[2]328[80,15]53[2]205[80,15]54[2,2]4010[80,16]22[2]328[80,16]23[2]328[80,16]53[2]205[80,16]54[2,2]4010[80,17]22[2]328[80,17]23[2]328[80,17]53[2]205 66

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Table 5-1 .Continued SmallGroupIDpBlockIDBrauerdegreesdimF(B)dimF(bas(B)) [80,17]54[2,2]4010[80,18]22[2]328[80,18]23[2]328[80,18]53[2]205[80,18]54[2,2]4010[80,19]55[2]205[80,19]56[2]205[80,29]53[2,2]4010[80,31]53[2,2]4010[80,33]53[2,2]4010[80,34]53[2,2]4010[80,35]55[2]205[80,35]56[2]205[80,37]55[2]205[80,37]56[2]205[80,38]55[2]205[80,38]56[2]205[80,39]22[2]328[80,39]23[2]328[80,39]55[2,2]4010[80,40]22[2]328[80,40]23[2]328[80,40]55[2,2]4010[80,41]22[2]328[80,41]23[2]328[80,41]55[2,2]4010[80,42]22[2]328[80,42]23[2]328[80,42]55[2,2]4010[80,44]55[2]205[80,44]56[2]205[84,5]33[2]123[84,5]34[2]123[84,5]35[2]123[84,5]36[2]123[84,5]37[2]123[84,5]38[2]123[84,5]73[2]287[84,5]74[2]287[84,8]33[2,2]246[84,8]34[2,2]246[84,8]35[2,2]246[84,8]73[2,2]5614[84,11]22[3]364 67

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Table 5-1 .Continued SmallGroupIDpBlockIDBrauerdegreesdimF(B)dimF(bas(B)) [84,11]23[3]364[84,11]72[3]637[84,14]33[2]123[84,14]34[2]123[84,14]35[2]123[84,14]36[2]123[84,14]37[2]123[84,14]38[2]123[84,14]73[2]287[84,14]74[2]287[88,3]113[2]4411[88,5]113[2]4411[88,7]113[2]4411[90,3]32[2]369[90,3]33[2]369[90,3]52[2]205[90,3]53[2]205[90,3]54[2]205[90,3]55[2]205[90,7]32[2]369[90,7]33[2]369[90,7]54[2]205[90,7]55[2]205[90,7]56[2]205[90,9]32[2]369[90,9]33[2]369[90,9]52[2]205[90,9]53[2]205[90,9]54[2]205[90,9]55[2]205 68

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BIOGRAPHICALSKETCH LeeRaneywasborninFortMyers,Florida.HegraduatedfromCypressLakeHighSchoolin2003.Heearnedabachelor'sdegreeinmathematicsfromFloridaAtlanticUniversityinBocaRaton,Floridain2006.HewentontopursuegraduatestudiesinmathematicsattheUniversityofFloridaandreceivedhismaster'sdegreeanddoctoratein2008and2013,respectively.Hisscholarlyinterestsinvolverepresentationtheoryofnitegroupsandmathematicseducation.Anavidfanofsports,heattendedmanyFloridaGatorsfootballandbasketballgamesduringhistimeattheUniversityofFlorida.Inhisfreetime,heenjoysultimatefrisbee,golf,videogames,listeningtomusic,andplayingtheguitar. 71