Groups St Andrews 2017 in Birmingham (London Mathematical Society Lecture Note Series, Band 455) PDF
Preview Groups St Andrews 2017 in Birmingham (London Mathematical Society Lecture Note Series, Band 455)
LONDONMATHEMATICALSOCIETYLECTURENOTESERIES ManagingEditor:ProfessorM.Reid,MathematicsInstitute, UniversityofWarwick,CoventryCV47AL,UnitedKingdom Thetitlesbelowareavailablefrombooksellers,orfromCambridgeUniversityPressat www.cambridge.org/mathematics 346 Surveysincombinatorics2007, A.HILTON&J.TALBOT(eds) 347 Surveysincontemporarymathematics, N.YOUNG&Y.CHOI(eds) 348 Transcendentaldynamicsandcomplexanalysis, P.J.RIPPON&G.M.STALLARD(eds) 349 ModeltheorywithapplicationstoalgebraandanalysisI, Z.CHATZIDAKIS,D.MACPHERSON, A.PILLAY&A.WILKIE(eds) 350 ModeltheorywithapplicationstoalgebraandanalysisII, Z.CHATZIDAKIS,D.MACPHERSON, A.PILLAY&A.WILKIE(eds) 351 FinitevonNeumannalgebrasandmasas, A.M.SINCLAIR&R.R.SMITH 352 Numbertheoryandpolynomials, J.MCKEE&C.SMYTH(eds) 353 Trendsinstochasticanalysis, J.BLATH,P.MÖRTERS&M.SCHEUTZOW(eds) 354 Groupsandanalysis, K.TENT(ed) 355 Non-equilibriumstatisticalmechanicsandturbulence, J.CARDY,G.FALKOVICH&K.GAWEDZKI 356 EllipticcurvesandbigGaloisrepresentations, D.DELBOURGO 357 Algebraictheoryofdifferentialequations, M.A.H.MACCALLUM&A.V.MIKHAILOV(eds) 358 Geometricandcohomologicalmethodsingrouptheory, M.R.BRIDSON,P.H.KROPHOLLER& I.J.LEARY(eds) 359 Modulispacesandvectorbundles, L.BRAMBILA-PAZ,S.B.BRADLOW,O.GARCÍA-PRADA& S.RAMANAN(eds) 360 Zariskigeometries, B.ZILBER 361 Words:Notesonverbalwidthingroups, D.SEGAL 362 Differentialtensoralgebrasandtheirmodulecategories, R.BAUTISTA,L.SALMERÓN&R.ZUAZUA 363 Foundationsofcomputationalmathematics,HongKong2008, F.CUCKER,A.PINKUS&M.J.TODD(eds) 364 Partialdifferentialequationsandfluidmechanics, J.C.ROBINSON&J.L.RODRIGO(eds) 365 Surveysincombinatorics2009, S.HUCZYNSKA,J.D.MITCHELL&C.M.RONEY-DOUGAL(eds) 366 Highlyoscillatoryproblems, B.ENGQUIST,A.FOKAS,E.HAIRER&A.ISERLES(eds) 367 Randommatrices:Highdimensionalphenomena, G.BLOWER 368 GeometryofRiemannsurfaces, F.P.GARDINER,G.GONZÁLEZ-DIEZ&C.KOUROUNIOTIS(eds) 369 Epidemicsandrumoursincomplexnetworks, M.DRAIEF&L.MASSOULIÉ 370 Theoryofp-adicdistributions, S.ALBEVERIO,A.YU.KHRENNIKOV&V.M.SHELKOVICH 371 Conformalfractals, F.PRZYTYCKI&M.URBAN´SKI 372 Moonshine:Thefirstquartercenturyandbeyond, J.LEPOWSKY,J.MCKAY&M.P.TUITE(eds) 373 Smoothness,regularityandcompleteintersection, J.MAJADAS&A.G.RODICIO 374 Geometricanalysisofhyperbolicdifferentialequations:Anintroduction, S.ALINHAC 375 Triangulatedcategories, T.HOLM,P.JØRGENSEN&R.ROUQUIER(eds) 376 Permutationpatterns, S.LINTON,N.RUŠKUC&V.VATTER(eds) 377 AnintroductiontoGaloiscohomologyanditsapplications, G.BERHUY 378 Probabilityandmathematicalgenetics, N.H.BINGHAM&C.M.GOLDIE(eds) 379 Finiteandalgorithmicmodeltheory, J.ESPARZA,C.MICHAUX&C.STEINHORN(eds) 380 Realandcomplexsingularities, M.MANOEL,M.C.ROMEROFUSTER&C.T.CWALL(eds) 381 Symmetriesandintegrabilityofdifferenceequations, D.LEVI,P.OLVER,Z.THOMOVA& P.WINTERNITZ(eds) 382 Forcingwithrandomvariablesandproofcomplexity, J.KRAJÍCˇEK 383 Motivicintegrationanditsinteractionswithmodeltheoryandnon-ArchimedeangeometryI, R.CLUCKERS, J.NICAISE&J.SEBAG(eds) 384 Motivicintegrationanditsinteractionswithmodeltheoryandnon-ArchimedeangeometryII, R.CLUCKERS, J.NICAISE&J.SEBAG(eds) 385 EntropyofhiddenMarkovprocessesandconnectionstodynamicalsystems, B.MARCUS,K.PETERSEN& T.WEISSMAN(eds) 386 Independence-friendlylogic, A.L.MANN,G.SANDU&M.SEVENSTER 387 GroupsStAndrews2009inBathI, C.M.CAMPBELLetal(eds) 388 GroupsStAndrews2009inBathII, C.M.CAMPBELLetal(eds) 389 Randomfieldsonthesphere, D.MARINUCCI&G.PECCATI 390 Localizationinperiodicpotentials, D.E.PELINOVSKY 391 Fusionsystemsinalgebraandtopology, M.ASCHBACHER,R.KESSAR&B.OLIVER 392 Surveysincombinatorics2011, R.CHAPMAN(ed) 393 Non-abelianfundamentalgroupsandIwasawatheory, J.COATESetal(eds) 394 Variationalproblemsindifferentialgeometry, R.BIELAWSKI,K.HOUSTON&M.SPEIGHT(eds) 395 Howgroupsgrow, A.MANN 396 Arithmeticdifferentialoperatorsoverthep-adicintegers, C.C.RALPH&S.R.SIMANCA 397 Hyperbolicgeometryandapplicationsinquantumchaosandcosmology, J.BOLTE&F.STEINER(eds) 398 Mathematicalmodelsincontactmechanics, M.SOFONEA&A.MATEI 399 Circuitdoublecoverofgraphs, C.-Q.ZHANG 400 Densespherepackings:ablueprintforformalproofs, T.HALES 401 AdoubleHallalgebraapproachtoaffinequantumSchur–Weyltheory, B.DENG,J.DU&Q.FU 402 Mathematicalaspectsoffluidmechanics, J.C.ROBINSON,J.L.RODRIGO&W.SADOWSKI(eds) 403 Foundationsofcomputationalmathematics,Budapest2011, F.CUCKER,T.KRICK,A.PINKUS& A.SZANTO(eds) 404 Operatormethodsforboundaryvalueproblems, S.HASSI,H.S.V.DESNOO&F.H.SZAFRANIEC(eds) 405 Torsors,étalehomotopyandapplicationstorationalpoints, A.N.SKOROBOGATOV(ed) 406 Appalachiansettheory, J.CUMMINGS&E.SCHIMMERLING(eds) 407 Themaximalsubgroupsofthelow-dimensionalfiniteclassicalgroups, J.N.BRAY,D.F.HOLT& C.M.RONEY-DOUGAL 408 Complexityscience:theWarwickmaster’scourse, R.BALL,V.KOLOKOLTSOV&R.S.MACKAY(eds) 409 Surveysincombinatorics2013, S.R.BLACKBURN,S.GERKE&M.WILDON(eds) 410 Representationtheoryandharmonicanalysisofwreathproductsoffinitegroups, T.CECCHERINI-SILBERSTEIN,F.SCARABOTTI&F.TOLLI 411 Modulispaces, L.BRAMBILA-PAZ,O.GARCÍA-PRADA,P.NEWSTEAD&R.P.THOMAS(eds) 412 Automorphismsandequivalencerelationsintopologicaldynamics, D.B.ELLIS&R.ELLIS 413 Optimaltransportation, Y.OLLIVIER,H.PAJOT&C.VILLANI(eds) 414 AutomorphicformsandGaloisrepresentationsI, F.DIAMOND,P.L.KASSAEI&M.KIM(eds) 415 AutomorphicformsandGaloisrepresentationsII, F.DIAMOND,P.L.KASSAEI&M.KIM(eds) 416 Reversibilityindynamicsandgrouptheory, A.G.O’FARRELL&I.SHORT 417 Recentadvancesinalgebraicgeometry, C.D.HACON,M.MUSTAT¸Aˇ&M.POPA(eds) 418 TheBloch–KatoconjecturefortheRiemannzetafunction, J.COATES,A.RAGHURAM,A.SAIKIA& R.SUJATHA(eds) 419 TheCauchyproblemfornon-Lipschitzsemi-linearparabolicpartialdifferentialequations, J.C.MEYER&D.J.NEEDHAM 420 Arithmeticandgeometry, L.DIEULEFAITetal(eds) 421 O-minimalityandDiophantinegeometry, G.O.JONES&A.J.WILKIE(eds) 422 GroupsStAndrews2013, C.M.CAMPBELLetal(eds) 423 Inequalitiesforgrapheigenvalues, Z.STANIC´ 424 Surveysincombinatorics2015, A.CZUMAJetal(eds) 425 Geometry,topologyanddynamicsinnegativecurvature, C.S.ARAVINDA,F.T.FARRELL& J.-F.LAFONT(eds) 426 Lecturesonthetheoryofwaterwaves, T.BRIDGES,M.GROVES&D.NICHOLLS(eds) 427 RecentadvancesinHodgetheory, M.KERR&G.PEARLSTEIN(eds) 428 GeometryinaFréchetcontext, C.T.J.DODSON,G.GALANIS&E.VASSILIOU 429 Sheavesandfunctionsmodulop, L.TAELMAN 430 RecentprogressinthetheoryoftheEulerandNavier–Stokesequations, J.C.ROBINSON, J.L.RODRIGO,W.SADOWSKI&A.VIDAL-LÓPEZ(eds) 431 Harmonicandsubharmonicfunctiontheoryontherealhyperbolicball, M.STOLL 432 Topicsingraphautomorphismsandreconstruction(2ndEdition), J.LAURI&R.SCAPELLATO 433 RegularandirregularholonomicD-modules, M.KASHIWARA&P.SCHAPIRA 434 Analyticsemigroupsandsemilinearinitialboundaryvalueproblems(2ndEdition), K.TAIRA 435 GradedringsandgradedGrothendieckgroups, R.HAZRAT 436 Groups,graphsandrandomwalks, T.CECCHERINI-SILBERSTEIN,M.SALVATORI& E.SAVA-HUSS(eds) 437 Dynamicsandanalyticnumbertheory, D.BADZIAHIN,A.GORODNIK&N.PEYERIMHOFF(eds) 438 Randomwalksandheatkernelsongraphs, M.T.BARLOW 439 Evolutionequations, K.AMMARI&S.GERBI(eds) 440 Surveysincombinatorics2017, A.CLAESSONetal(eds) 441 Polynomialsandthemod2SteenrodalgebraI, G.WALKER&R.M.W.WOOD 442 Polynomialsandthemod2SteenrodalgebraII, G.WALKER&R.M.W.WOOD 443 Asymptoticanalysisingeneralrelativity, T.DAUDÉ,D.HÄFNER&J.-P.NICOLAS(eds) 444 Geometricandcohomologicalgrouptheory, P.H.KROPHOLLER,I.J.LEARY,C.MARTÍNEZ-PÉREZ& B.E.A.NUCINKIS(eds) 445 Introductiontohiddensemi-Markovmodels, J.VANDERHOEK&R.J.ELLIOTT 446 Advancesintwo-dimensionalhomotopyandcombinatorialgrouptheory, W.METZLER& S.ROSEBROCK(eds) 447 Newdirectionsinlocallycompactgroups, P.-E.CAPRACE&N.MONOD(eds) 448 Syntheticdifferentialtopology, M.C.BUNGE,F.GAGO&A.M.SANLUIS 449 Permutationgroupsandcartesiandecompositions, C.E.PRAEGER&C.SCHNEIDER 450 Partialdifferentialequationsarisingfromphysicsandgeometry, M.BENAYEDetal(eds) 451 Topologicalmethodsingrouptheory, N.BROADDUS,M.DAVIS,J.-F.LAFONT&I.ORTIZ(eds) 452 Partialdifferentialequationsinfluidmechanics, C.L.FEFFERMAN,J.C.ROBINSON& J.L.RODRIGO(eds) 453 Stochasticstabilityofdifferentialequationsinabstractspaces, K.LIU 454 Beyondhyperbolicity, M.HAGEN,R.WEBB&H.WILTON(eds) LondonMathematicalSocietyLectureNoteSeries:455 Groups St Andrews 2017 in Birmingham Editedby C.M.CAMPBELL UniversityofStAndrews C.W.PARKER UniversityofBirmingham M.R.QUICK UniversityofStAndrews E.F.ROBERTSON UniversityofStAndrews C.M.RONEY-DOUGAL UniversityofStAndrews UniversityPrintingHouse,CambridgeCB28BS,UnitedKingdom OneLibertyPlaza,20thFloor,NewYork,NY10006,USA 477WilliamstownRoad,PortMelbourne,VIC3207,Australia 314–321,3rdFloor,Plot3,SplendorForum,JasolaDistrictCentre,NewDelhi–110025,India 79AnsonRoad,#06-04/06,Singapore079906 CambridgeUniversityPressispartoftheUniversityofCambridge. ItfurtherstheUniversity’smissionbydisseminatingknowledgeinthepursuitof education,learning,andresearchatthehighestinternationallevelsofexcellence. www.cambridge.org Informationonthistitle:www.cambridge.org/9781108728744 DOI:10.1017/9781108692397 ©CambridgeUniversityPress2019 Thispublicationisincopyright.Subjecttostatutoryexception andtotheprovisionsofrelevantcollectivelicensingagreements, noreproductionofanypartmaytakeplacewithoutthewritten permissionofCambridgeUniversityPress. Firstpublished2019 PrintedandboundinGreatBritainbyClaysLtd,ElcografS.p.A. AcataloguerecordforthispublicationisavailablefromtheBritishLibrary. LibraryofCongressCataloging-in-PublicationData Names:GroupsSt.Andrews(Conference)(2017:UniversityofBirmingham)| Campbell,C.M.,1942-editor. Title:GroupsSt.Andrews2017inBirmingham/editedbyC.M.Campbell[and fourothers]. Description:Cambridge;NewYork,NY:CambridgeUniversityPress,[2019]| Series:LondonMathematicalSocietylecturenoteseries;455|Includes bibliographicalreferences. Identifiers:LCCN2018058651|ISBN9781108728744(hardback:alk.paper) Subjects:LCSH:Grouptheory--Congresses.|Algebra--Congresses. Classification:LCCQA174.G77442017|DDC512/.2--dc23 LCrecordavailableathttps://lccn.loc.gov/2018058651 ISBN978-1-108-72874-4Paperback CambridgeUniversityPresshasnoresponsibilityforthepersistenceoraccuracyof URLsforexternalorthird-partyinternetwebsitesreferredtointhispublication anddoesnotguaranteethatanycontentonsuchwebsitesis,orwillremain, accurateorappropriate. CONTENTS Introduction..................................................................vii Finite simple groups and fusion systems Michael Aschbacher............................................................1 Finite and infinite quotients of discrete and indiscrete groups Pierre-Emmanuel Caprace ....................................................16 Local-global conjectures and blocks of finite simple groups Radha Kessar & Gunter Malle................................................70 A survey on some methods of generating finite simple groups Ayoub B.M. Basheer & Jamshid Moori......................................106 One-relator groups: an overview Gilbert Baumslag, Benjamin Fine & Gerhard Rosenberger....................119 New progress in products of conjugacy classes in finite groups Antonio Beltr´an, Mar´ıa Jos´e Felipe & Carmen Melchor......................158 Aspherical relative presentations all over again William A. Bogley, Martin Edjvet & Gerald Williams........................169 Simple groups, generation and probabilistic methods Timothy C. Burness.........................................................200 Irreducible subgroups of simple algebraic groups – a survey Timothy C. Burness & Donna M. Testerman ................................230 Practical computation with linear groups over infinite domains A.S. Detinko & D.L. Flannery..............................................261 Beauville p-groups: a survey Ben Fairbairn................................................................271 Structural criteria in factorised groups via conjugacy class sizes Mar´ıa Jos´e Felipe, Ana Mart´ınez-Pastor & V´ıctor Manuel Ortiz-Sotomayor..289 Growth in linear algebraic groups and permutation groups: towards a unified perspective Harald A. Helfgott...........................................................300 vi L2-Betti numbers and their analogues in positive characteristic Andrei Jaikin-Zapirain.......................................................346 On the pronormality of subgroups of odd index in finite simple groups Anatoly S. Kondrat’ev, Natalia Maslova & Danila Revin.....................406 Vertex stabilizers of graphs with primitive automorphism groups and a strong version of the Sims conjecture Anatoly S. Kondrat’ev & Vladimir I. Trofimov...............................419 On the character degrees of a Sylow p-subgroup of a finite Chevalley group G(pf) over a bad prime Tung Le, Kay Magaard & Alessandro Paolini ................................427 Patterns on symmetric Riemann surfaces Adnan Meleko˘glu & David Singerman........................................443 Subgroups of twisted wreath products P´eter P. Pa´lfy...............................................................455 Some remarks on self-dual codes invariant under almost simple permutation groups B.G. Rodrigues & T.M. Mudziiri Shumba ...................................469 Test elements: from pro-p to discrete groups Ilir Snopce & Slobodan Tanushevski..........................................486 INTRODUCTION GroupsStAndrews2017washeldattheUniversityofBirminghamfrom5thAu- gust to 13th August 2017. This was the tenth in the series of Groups St Andrews group theory conferences. There were over 200 mathematicians involved in the meeting as well as some family members and partners. The Scientific Organising Committee of Groups St Andrews 2017 was Colin Campbell, Martyn Quick, Ed- mundRobertsonandColvaRoney-Dougal(allfromStAndrews)andChrisParker (Birmingham). The academic business of the conference ran for seven days from Sunday 6th August to Saturday 12th August. Four main speakers delivered four talks each, surveying areas of contemporary development in group theory and related areas: Michael Aschbacher (Caltech), Pierre-Emmanuel Caprace (Universit´e Catholique de Louvain), Radha Kessar (City, University of London) and Gunter Malle (TU Kaiserslautern). Therewerefiveinvitedspeakersdeliveringone-hourplenarytalks: Tim Burness (University of Bristol), Vincent Guirardel (Universit´e de Rennes 1), Harald Helfgott (University of G¨ottingen), Andrei Jaikin-Zapirain (Universidad Auto´noma de Madrid) and Donna Testerman (E´cole Polytechnique F´ed´erale de Lausanne). In addition there were about 115 contributed short talks from the delegates. In the evenings throughout the conference there was an extensive social pro- gramme. The main conference outing was a choice between Stratford upon Avon (with a chance to visit buildings and sites associated with William Shakespeare) and WarwickCastle. Other highlightsof thesocial programmewereawinerecep- tion,amusicaleveningandtheconferencedinner. OnceagainT˙DailyGroupT˙ori< was a nice feature of the conference. We thank the various editors of this, by now traditional, publication. Thesupport of thetwomain United Kingdommathematicssocieties, the Edin- burghMathematicalSocietyandtheLondonMathematicalSocietyhas,onceagain, beenanimportantfactorinthesuccessoftheseconferences. Aswellassupporting someoftheexpensesofthemainspeakers,thegrantsfromthesesocietieswereused to support postgraduate students and also participants from Scheme 5 countries. FinancialsupportwasalsoreceivedfromtheHeilbronnInstituteforMathematical Research and the publishers de Gruyter, Elsevier and Springer Nature. Onceagainallthemainspeakershavewrittensubstantialsurveys,inonecasea joint article, for these Proceedings. The other papers are also of a survey nature. We would like to thank Martyn Quick and Colva Roney-Dougal not only for their editorial assistance with these Proceedings but also for all their hard work in organising the conference. Last, and by no means least, we would like to thank ChrisParker,thelocalhero,forallhisuntiringeffortsinmakingBirminghamsuch a happy venue. CMC, EFR FINITE SIMPLE GROUPS AND FUSION SYSTEMS MICHAEL ASCHBACHER CaliforniaInstituteofTechnology,Pasadena,California91125,USA Email: [email protected] Thisexpositorypaperistakenfromaseriesoffourtalksgivenattheconference Groups St Andrews in Birmingham 2017, held in August of 2017. The goal of thosetalkswastogivetheaudiencesomeinsightintoanongoingprogramto,first, classify a certain class of simple 2-fusion systems, and then, second, to use the result on fusion systems to simplify the proof of the theorem classifying the finite simple groups (CFSG). But since the talks were delivered to a general audience ofgrouptheorists,mostofthepresentationwasdevotedtosupplyingbackground. The same is true of this article, where the program does not formally make an appearance until fairly late in the game. Thus we’ll begin with an introduction to the basic theory of fusion systems. Then we give an overview of the proof of that part of the CFSG devoted to the groupsofcomponenttype, afterwhichwediscusshowtotranslatethatproofinto the category of 2-fusion systems, and indicate some advantages that accrue from that translation. We also describe some other changes to the original proof of the CFSG that are part of the program. Our basic reference on fusion systems is [2], although [7] also supplies a good introductiontothesubject. Ourbasicreferenceonfinitegroupsis[1]. Foramore detailed discussion of the proof of the CFSG see [3]. Fusion systems LetpbeaprimeandSafinitep-group. AfusionsystemonSisacategoryF whose objects are the subgroups of S and, for subgroups P,Q of S, the set homF(P,Q) ofmorphismsfromP toQisasetofinjectivegrouphomomorphismsofP intoQ, and that set satisfies two weak axioms: (1) If s∈S with Ps ≤Q then the conjugation map c :P →Q is a morphism. s (2) If φ:P →Q is a morphism, then so is φ:P →Pφ and φ−1 :Pφ→P. Call S the Sylow group of F. Example 1.1 LetGbeafinitegroup,S ∈Syl (G),andF (G)thefusionsystem p S onS whosemorphismsareinducedviaconjugationinG. CallF (G)thep-fusion S system of G. We are primarily interested in saturated fusion systems. A fusion system F is saturated if it satisfies two more axioms, that can be easily seen to hold in Example 1.1 using Sylow’s Theorem. See [2] for the axioms. ThisworkwaspartiallysupportedbyDMSNSF-1265587andDMSNSF-1601063.
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