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OXFORD MATHEMATICAL MONOGRAPHS Series Editors J. M. BALL E. M. FRIEDLANDER I. G. MACDONALD L. NIRENBERG R. PENROSE J. T. STUART N. J. HITCHIN W. T. GOWERS OXFORD MATHEMATICAL MONOGRAPHS L.Ambrosio,N.FuscoandD.Pallara:Functionsofboundedvariationandfreediscontinuityproblems A.Belleni-Moranti:Appliedsemigroupsandevolutionequations A.M.Arthurs:Complementaryvariationalprinciples 2ndedition M.RosenblumandJ.Rovnyak:Hardyclassesandoperatortheory J.W.P.Hirschfeld:Finiteprotectivespacesofthreedimensions A.PressleyandG.Segal:Loopgroups D.E.EdmundsandW.D.Evans:Spectraltheoryanddifferentialoperators WangJianhua:Thetheoryofgames S.OmatuandJ.H.Seinfeld:Distributedparametersystems:theoryandapplications J.Hilgert,K.H.Hofmann,andJ.D.Lawson:Liegroups,convexcones,andsemigroups S.Dineen:TheSchwarzlemma S.K.DonaldsonandP.R.Kronheimer:Thegeometryoffour-manifolds D.W.Robinson:EllipticoperatorsandLiegroups A.G.Werschulz:Thecomputationalcomplexityofdifferentialandintegralequations L.Evens:Cohomologyofgroups G.EffingerandD.R.Hayes:Additivenumbertheoryofpolynomials J.W.P.HirschfeldandJ.A.Thas:GeneralGaloisgeometries P.N.HoffmanandJ.F.Humphreys:Protectiverepresentationsofthesymmetricgroups I.Gyo¨riandG.Ladas:Theoscillationtheoryofdelaydifferentialequations J.Heinonen,T.Kilpelainen,andO.Martio:Non-linearpotentialtheory B.Amberg,S.Franciosi,andF.deGiovanni:Productsofgroups M.E.Gurtin:Thermomechanicsofevolvingphaseboundariesintheplane I.IonescuandM.Sofonea:Functionalandnumericalmethodsinviscoplasticity N.Woodhouse:Geometricquantization 2ndedition U.Grenander:Generalpatterntheory J.FarautandA.Koranyi:Analysisonsymmetriccones I.G.Macdonald:SymmetricfunctionsandHallpolynomials2ndedition B.L.R.ShawyerandB.B.Watson:Borel’smethodsofsummability M.Holschneider:Wavelets:ananalysistool JacquesThévenaz:G-algebrasandmodularrepresentationtheory Hans-JoachimBaues:Homotopytypeandhomology P.D.D’Eath:Blackholes:gravitationalinteractions R.Lowen:Approachspaces:themissinglinkinthetopology–uniformity–metrictriad NguyenDinhCong:Topologicaldynamicsofrandomdynamicalsystems J.W.P.Hirschfeld:Protectivegeometriesoverfinitefields 2ndedition K.MatsuzakiandM.Taniguchi:HyperbolicmanifoldsandKleiniangroups DavidE.EvansandYasuyukiKawahigashi:Quantumsymmetriesonoperatoralgebras NorbertKlingen:Arithmeticalsimilarities:primedecompositionandfinitegrouptheory IsabelleCatto,ClaudeLeBris,andPierre-LouisLions:Themathematicaltheoryofthermodynamic limits:Thomas–Fermitypemodels D.McDuffandD.Salamon:Introductiontosymplectictopology 2ndedition WilliamM.Goldman:Complexhyperbolicgeometry CharlesJ.ColbournandAlexanderRosa:Triplesystems V.A.Kozlov,V.G.Maz’ya,andA.B.Movchan:Asymptoticanalysisoffieldsinmulti-structures GérardA.Maugin:Nonlinearwavesinelasticcrystals GeorgeDassiosandRalphKleinman:Lowfrequencyscattering GeraldW.JohnsonandMichelL.Lapidus:TheFeynmanIntegralandFeynman’sOperationalCalculus W.LayandS.Y.Slavyanov:SpecialFunctions:AUnifiedtheorybasedonsingularities D.Joyce:CompactManifoldswithSpecialHolonomy A.CarboneandS.Semmes:Agraphicapologyforsymmetryandimplicitness JohannBoos:Classicalandmodernmethodsinsummability NigelHigsonandJohnRoe:AnalyticK-Homology S.Semmes:Somenoveltypesoffractalgeometry TadeuszIwaniecandGavenMartin:Geometricfunctiontheoryandnonlinearanalysis TerryLyonsandZhongminQian:Systemcontrolandroughpaths AndrewRanicki:Algebraicandgeometricsurgery LeonEhrenpreis:Theuniversalityoftheradontransform JohnC.LennoxandDerekJ.S.Robinson:Thetheoryofinfinitesolublegroups A.A.Ivanov:TheFourthJankoGroup The Fourth Janko Group A. A. IVANOV CLARENDON PRESS • OXFORD 2004 1 GreatClarendonStreet,OxfordOX26DP OxfordUniversityPressisadepartmentoftheUniversityofOxford. ItfurtherstheUniversity’sobjectiveofexcellenceinresearch,scholarship, andeducationbypublishingworldwidein Oxford NewYork Auckland Bangkok BuenosAires CapeTown Chennai DaresSalaam Delhi HongKong Istanbul Karachi Kolkata KualaLumpur Madrid Melbourne MexicoCity Mumbai Nairobi SãoPaulo Shanghai Taipei Tokyo Toronto OxfordisaregisteredtrademarkofOxfordUniversityPress intheUKandincertainothercountries PublishedintheUnitedStates byOxfordUniversityPressInc.,NewYork (cid:1)c OxfordUniversityPress,2004 Themoralrightsoftheauthorhavebeenasserted DatabaserightOxfordUniversityPress(maker) Firstpublished2004 Allrightsreserved.Nopartofthispublicationmaybereproduced, storedinaretrievalsystem,ortransmitted,inanyformorbyanymeans, withoutthepriorpermissioninwritingofOxfordUniversityPress, orasexpresslypermittedbylaw,orundertermsagreedwiththeappropriate reprographicsrightsorganization.Enquiriesconcerningreproduction outsidethescopeoftheaboveshouldbesenttotheRightsDepartment, OxfordUniversityPress,attheaddressabove Youmustnotcirculatethisbookinanyotherbindingorcover andyoumustimposethissameconditiononanyacquirer BritishLibraryCataloguinginPublicationData (Dataavailable) LibraryofCongressCataloginginPublicationData (Dataavailable) ISBN 0-19-852759-4 1 3 5 7 9 10 8 6 4 2 TypesetbyNewgenImagingSystems(P)Ltd.,Chennai,India PrintedinGreatBritain onacid-freepaperby BiddlesLtd.,King’sLynn,Norfolk To Lena -Ïîçâîëü,ñóäàðûíÿ,ìíåñäåëàòüòîæåòî÷íî, Â÷åìóïðàæíÿëèñüòå,êòîäåëàëèòåáÿ, Àâîñüèìíåóäàñòñÿíåíàðî÷íî Ñäåëàòüòàêóþæ,õîòüíåäëÿñåáÿ. Èâ íÁ êîâ“Òðåáîâàíèå” A AP PREFACE Westartwiththefollowingclassicalsituation. LetV bea10-dimensionalvector space over the field GF(2) of two elements. Let q be a non-singular quadratic ∼ + form of Witt index 5 on V. Let H = O (2) be the group of invertible linear 10 transformations of V that preserve q. Let Ω = D+(10,2) be the dual polar graph associated with the pair (V,q), that is the graph on the set of maximal subspaces of V which are totally singular with respect to q (these subspaces are 5-dimensional); two such subspaces are adjacent in Ω if and only if their intersection is of codimension 1 in each of the two subspaces. Then (Ω,H) = (D+(10,2),O+(2)) belongs to the class of pairs (Ξ,X), where Ξ is a graph and 10 X is a group of automorphisms of Ξ satisfying the following conditions (C1) to (C3): (C1) Ξ is connected of valency 31=25−1; (C2) the group X acts transitively on the set of incident vertex-edge pairs in Ξ; (C3) the stabilizer in X of a vertex of Ξ is the semi-direct product with respect to the natural action of the general linear group in dimension 5 over the field of two elements and the exterior square of the natural module of the linear group. The constrains imposed by the conditions (C1) to (C3) concern only ‘local’ properties of the action of X on Ξ and these properties remain unchanged when onetakessuitable‘coverings’. Theconstrainsareencodedinthestructureofthe stabilizers in X of a vertex of Ξ and of an edge containing this vertex, and also in the way these two stabilizers intersect. Denote the vertex and edge stabilizers by X[0] and X[1], respectively and ‘cut them out’ of X to obtain what is called the amalgam X ={X[0],X[1]} (theunionoftheelement-setsofthetwogroupswithgroupoperationscoinciding on the intersection X[01] = X[0]∩X[1]). Because of (C2) the isomorphism type of X is independent of the choice of the incident vertex–edge pair. If the pair (Ξ,X)issimplyconnectedwhichmeansthatΞisatree, thenX istheuniversal completion of X which is known to be the free product of X[0] and X[1] amal- gamatedoverthecommonsubgroupX[01].Areasonablequestiontoaskisabout the possibilities for the isomorphism type of X. The following lemma gives the answer. Lemma A. Let (Ξ,X) be a pair satisfying (C1) to (C3) and let X be the amalgam formed by the stabilizers in X of a vertex of Ξ and of an edge incident to vii viii Preface this vertex. Then X is either the classical amalgam H contained in H =O+(2) 10 or one extra amalgam G ={G[0],G[1]}. In a certain sense the existence of the additional amalgam G in Lemma A is due to the famous isomorphism of the general linear group in dimension four over the field of two elements and the alternating group of degree eight. Inordertoobtainallthepairssatisfying(C1)to(C3)weshouldconsiderthe quotients of the universal completions of the amalgams H and G over suitable normal subgroups. However, there are far too many possibilities for choosing thatnormalsubgroupandtheprojectoffindingthemallappearsfairlyhopeless. Nevertheless, we may still try to find particular examples which are ‘small and nice’ in one sense or another. It can be shown that for every pair (Ξ,X) satisfying (C1) to (C3) there is a ‘nice’ family of cubic (that is valency 3) subgraphs in Ξ as described in the following lemma. Lemma B. Let (Ξ,X) be a pair satisfying (C1) to (C3). Then Ξ contains a family S of connected subgraphs of valency 3. This family is unique subject to the condition that it is stabilized by X and whenever two subgraphs from S share a vertex and if the neighbours of this vertex in both subgraphs coincide, the whole graphs are equal. The subgraphs forming the family S in Lemma B are called geometric cubic subgraphs. If Ξ is a tree then every geometric cubic subgraph is a cubic tree which is ‘large’, even infinite. On the other hand, in the classical example (D+(10,2),O+(2)) the geometric cubic subgraphs correspond to the 10 3-dimensional totally singular subspaces in the 10-dimensional orthogonal space V. The subgraph corresponding to such a subspace U is formed by the maximal totally singular subspaces in V containing U. This subgraph is complete bipartite on 6 vertices denoted by K3,3. Thus here the geometric cubic subgraphs are small and nice. Let (Ξ,X) be a pair satisfying (C1) to (C3). Let Σ be a geometric cubic subgraph in Ξ, let S and T be the global and the vertexwise stabilizers of Σ in (cid:1) X. Let Σ be the graph on the set of orbits of the centralizer C (T) of T in S on S the vertex set of Σ in which two orbits are adjacent if there is at least one edge of Σ which joins them. Then the natural mapping ψ :Σ→Σ(cid:1) turns out to be a covering of graphs commuting with the action of S. Put (cid:1) S =S/(TC (T)) S which is the image of S in the outer automorphism group of T. Direct but somewhat tricky calculation in the amalgams H and G give the following. Preface ix Lemma C. In the above terms the following hold: (i) if X = H then S(cid:1) ∼= Sym (cid:4)Sym and Σ(cid:1) is the complete bipartite graph 3 2 K3,3: (cid:1) (cid:1) (cid:1) (cid:1)(cid:3) (cid:2)(cid:1) (cid:4)(cid:2) (cid:3) (cid:4) (cid:1) (cid:3)(cid:2) (cid:1)(cid:4) (cid:2) (cid:1)(cid:2) (cid:3)(cid:4) (cid:1)(cid:2) (cid:2)(cid:1)(cid:4) (cid:3)(cid:2)(cid:1) (cid:4) (cid:3) (cid:2)(cid:4) (cid:1) (cid:2) (cid:3)(cid:1) (cid:2)(cid:1)(cid:4) (cid:1)(cid:1)(cid:2) (cid:3)(cid:1)(cid:1) (ii) if X =G then S(cid:1)∼=Sym and Σ(cid:1) is the Petersen graph: 5 (cid:1) (cid:1) (cid:6)(cid:1) (cid:6) (cid:5)(cid:5)(cid:1) (cid:6)(cid:1) (cid:5)(cid:1) (cid:1) (cid:1) (cid:7) (cid:8) (cid:7) (cid:8) (cid:1)(cid:7) (cid:8)(cid:1) By the remark after Lemma B if the pair (Ξ,X) is (D+(10,2),O+(2)) then 10 the mapping ψ : Σ → Σ(cid:1) is an isomorphism. The following characterization has been established by P. J. Cameron and C. E. Praeger in 1982. Proposition D. Let (Ξ,X) be a pair satisfying (C1) to (C3) with X =H and suppose that ψ :Σ→Σ(cid:1) is an isomorphism. Then (Ξ,X)=(D+(10,2),O+(2)). 10 ItisnaturaltoaskwhathappenswhenΣattainstheotherminimalpossibility in Lemma C, that is the Petersen graph. The main purpose of this book is to answer this questing by proving the following Main Theorem. Let (Ξ,X) be a pair satisfying (C1) to (C3) with X =G and suppose that ψ :Σ→Σ(cid:1) is an isomorphism. Then the pair (Ξ,X) is determined uniquely up to isomorphism. Furthermore (i) X is non-abelian simple; (ii) |X|=221·33·5·7·113·23·29·31·37·43= 86,775,571,046,077,562,880; (iii) X contains an involution z such that CX(z)∼=21++12·3·Aut (M22). ThegroupX intheMainTheoremisthefourthJankosporadicsimplegroup J4 discovered by Zvonimir Janko in 1976 and constructed in Cambridge in 1980 by D. J. Benson, J. H. Conway, S. P. Norton, R. A Parker, J. G. Thackray as a subgroup of L112(2). x Preface The book was derived from research which originated almost exactly twenty years ago. This was the golden age of the research seminar on algebra and geometry at the Institute for System Studies (VNIISI) in Moscow. Some of its members were examining a special case of the quasithin groups’ classification (and J4 is a quasithin group); others were searching for new distance-transitive graphs. The latter activity led me to the discovery of the geometry F(J4). At this stage our quasithin experts suggested that the discovery might eventually emerge into a geometric construction and uniqueness proof for J4. The project that seemed preposterous at that time has been fully realized here. My foremost gratitude goes to Vladimir L’vovich Arlazarov and Igor AlexandrovichFaradjev,whobothfoundedandmaintainedtheVNIISIseminar. The crucial ingredient in the characterization is the simple connectedness of the geometry F(J4) which I proved in the summer of 1989 within the joint project withSergeyShpectorovontheclassificationofthePetersenandtildegeometries. I am very thankful to Sergey for his long term friendship and cooperation and hope this will last. When the proof was discussed in (then still West) Berlin in June1990,GeoffMasonnoticedthatthesimpleconnectednessformedabasisfor the uniqueness proof for J4. At the Durham Symposium in July 1990, together with Ulrich Meierfrankenfeld, we found a way of transferring the simple connectedness into a computer-free construction. It took a further ten years to see the construction published. Many original ideas here and indeed almost the whole of chapter 8 are due to Ulrich. The main layout of the book was designed when I gave a series of lectures at the University of Tokyo in autumn 2002. I am very thankful to Atsushi Matsuo whoorganizedtheselecturesandtotheaudienceformanystimulatingquestions anddiscussions.MyspecialthanksgotoSatoshiYoshiaraandHirokiShimakura who took the notes of the lectures; I used these notes as a draft for the first two chapters of the book. Hiroki wrote the notes in Japanese and now they are published as (Ivanov 2003). In order to proceed with the project I needed a transparent description of the pentad subgroup in J4. This was achieved in Oberwolfach in summer of 2003, again, thanks to a fruitful cooperation with Sergey Shpectorov. The writing up began in September 2003 and took about six months. Half way through I received useful comments on the draft from Ernie Shult. His note: ‘nice’ against Lemma 4.5.5 was particularly inspiring. Thankyou,Ernie.CorinnaWiedornisoneoftheveryfewwhoreadthroughour construction paper with Ulrich. So even during the early stages of working on the book, I was sure that it will have at least one dedicated reader. Her incisive comments on the final draft exceeded my best expectations. Antonio Pasini not only kindly offered to read the final draft but also assured me that this was not a favour at all and that this was solely for his own pleasure. His thoughtful comments are of great value to me. IampleasedtoacknowledgethatmytheinsightinJ4 wasgraduallybuiltup through discussions with experts including Michael Aschbacher, John Conway, Wolfgang Lempken, Simon Norton, Richard Parker, Gernot Stroth and Richard Weiss. Whenever I got stuck, Dima Pasechnik was always ready to help