LONDONMATHEMATICALSOCIETYLECTURENOTESERIES ManagingEditor: ProfessorN.J.Hitchin,MathematicalInstitute,UniversityofOxford,24-29StGiles,Oxford OXI3LB,UnitedKingdom Thetitlesbelowareavailablefrombooksellers,orfromCambridgeUniversityPressat www.cambridge.org/mathematics 214 GeneralisedEuler-Jacobiinversionformulaandasymptoticsbeyondallorders,V.KOWALENKOetal 215 Numbertheory1992–93,S.DAVID(ed) 216 Stochasticpartialdifferentialequations,A.ETHERIDGE(ed) 217 Quadraticformswithapplicationstoalgebraicgeometryandtopology,A.PFISTER 218 Surveysincombinatorics,1995,P.ROWLINSON(ed) 220 Algebraicsettheory,A.JOYAL&I.MOERDIJK 221 Harmonicapproximation.,S.J.GARDINER 222 Advancesinlinearlogic,J.-Y.GIRARD,Y.LAFONT&L.REGNIER(eds) 223 Analyticsemigroupsandsemilinearinitialboundaryvalueproblems,KAZUAKITAIRA 224 Computability,enumerability,unsolvability,S.B.COOPER,T.A.SLAMAN& S.S.WAINER(eds) 225 Amathematicalintroductiontostringtheory,S.ALBEVERIO,etal 226 Novikovconjectures,indextheoremsandrigidityI,S.FERRY,A.RANICKI&J.ROSENBERG(eds) 227 Novikovconjectures,indextheoremsandrigidityII,S.FERRY,A.RANICKI&J.ROSENBERG(eds) 228 ErgodictheoryofZdactions,M.POLLICOTT&K.SCHMIDT(eds) 229 Ergodicityforinfinitedimensionalsystems,G.DAPRATO&J.ZABCZYK 230 Prolegomenatoamiddlebrowarithmeticofcurvesofgenus2,J.W.S.CASSELS&E.V.FLYNN 231 Semigrouptheoryanditsapplications,K.H.HOFMANN&M.W.MISLOVE(eds) 232 ThedescriptivesettheoryofPolishgroupactions,H.BECKER&A.S.KECHRIS 233 Finitefieldsandapplications,S.COHEN&H.NIEDERREITER(eds) 234 Introductiontosubfactors,V.JONES&V.S.SUNDER 235 Numbertheory1993–94,S.DAVID(ed) 236 TheJamesforest,H.FETTER&B.G.DEBUEN 237 Sievemethods,exponentialsums,andtheirapplicationsinnumbertheory, G.R.H.GREAVESetal 238 Representationtheoryandalgebraicgeometry,A.MARTSINKOVSKY&G.TODOROV(eds) 240 Stablegroups,F.O.WAGNER 241 Surveysincombinatorics,1997,R.A.BAILEY(ed) 242 GeometricGaloisactionsI,L.SCHNEPS&P.LOCHAK(eds) 243 GeometricGaloisactionsII,L.SCHNEPS&P.LOCHAK(eds) 244 Modeltheoryofgroupsandautomorphismgroups,D.EVANS(ed) 245 Geometry,combinatorialdesignsandrelatedstructures,J.W.P.HIRSCHFELDetal 246 p-Automorphismsoffinitep-groups,E.I.KHUKHRO 247 Analyticnumbertheory,Y.MOTOHASHI(ed) 248 Tametopologyando-minimalstructures,L.VANDENDRIES 249 Theatlasoffinitegroups: tenyearson,R.CURTIS&R.WILSON(eds) 250 Charactersandblocksoffinitegroups,G.NAVARRO 251 Gr¨obnerbasesandapplications,B.BUCHBERGER&F.WINKLER(eds) 252 Geometryandcohomologyingrouptheory,P.KROPHOLLER,G.NIBLO&R.STO¨HR(eds) 253 Theq-Schuralgebra,S.DONKIN 254 Galoisrepresentationsinarithmeticalgebraicgeometry,A.J.SCHOLL&R.L.TAYLOR(eds) 255 Symmetriesandintegrabilityofdifferenceequations,P.A.CLARKSON&F.W.NIJHOFF(eds) 256 AspectsofGaloistheory,H.VO¨LKLEINetal 257 Anintroductiontononcommutativedifferentialgeometryanditsphysicalapplications2ed, J.MADORE 258 Setsandproofs,S.B.COOPER&J.TRUSS(eds) 259 Modelsandcomputability,S.B.COOPER&J.TRUSS(eds) 260 GroupsStAndrews1997inBath,I,C.M.CAMPBELLetal 261 GroupsStAndrews1997inBath,II,C.M.CAMPBELLetal 262 Analysisandlogic,C.W.HENSON,J.IOVINO,A.S.KECHRIS& E.ODELL 263 Singularitytheory,B.BRUCE&D.MOND(eds) 264 Newtrendsinalgebraicgeometry,K.HULEK,F.CATANESE,C.PETERS&M.REID(eds) 265 Ellipticcurvesincryptography,I.BLAKE,G.SEROUSSI&N.SMART 267 Surveysincombinatorics,1999,J.D.LAMB&D.A.PREECE(eds) 268 Spectralasymptoticsinthesemi-classicallimit,M.DIMASSI&J.SJO¨STRAND 269 Ergodictheoryandtopologicaldynamics,M.B.BEKKA&M.MAYER 270 AnalysisonLiegroups,N.T.VAROPOULOS&S.MUSTAPHA 271 Singularperturbationsofdifferentialoperators,S.ALBEVERIO&P.KURASOV 272 Charactertheoryfortheoddordertheorem,T.PETERFALVI 273 Spectraltheoryandgeometry,E.B.DAVIES&Y.SAFAROV(eds) 274 TheMandlebrotset,themeandvariations,TANLEI(ed) 275 Descriptivesettheoryanddynamicalsystems,M.FOREMANetal 276 Singularitiesofplanecurves,E.CASAS-ALVERO 277 Computationalandgeometricaspectsofmodernalgebra,M.D.ATKINSONetal 278 Globalattractorsinabstractparabolicproblems,J.W.CHOLEWA&T.DLOTKO 279 Topicsinsymbolicdynamicsandapplications,F.BLANCHARD,A.MAASS&A.NOGUEIRA(eds) 280 CharactersandautomorphismgroupsofcompactRiemannsurfaces,T.BREUER 281 Explicitbirationalgeometryof3-folds,A.CORTI&M.REID(eds) 282 Auslander-Buchweitzapproximationsofequivariantmodules,M.HASHIMOTO 283 Nonlinearelasticity,Y.FU&R.W.OGDEN(eds) 284 Foundationsofcomputationalmathematics,R.DEVORE,A.ISERLES&E.SU¨LI(eds) 285 Rationalpointsoncurvesoverfinite,fields,H.NIEDERREITER&C.XING 286 Cliffordalgebrasandspinors2ed,P.LOUNESTO 287 TopicsonRiemannsurfacesandFuchsiangroups,E.BUJALANCEetal 288 Surveysincombinatorics,2001,J.HIRSCHFELD(ed) 289 AspectsofSobolev-typeinequalities,L.SALOFF-COSTE 290 QuantumgroupsandLietheory,A.PRESSLEY(ed) 291 Titsbuildingsandthemodeltheoryofgroups,K.TENT(ed) 292 Aquantumgroupsprimer,S.MAJID 293 SecondorderpartialdifferentialequationsinHilbertspaces,G.DAPRATO&J.ZABCZYK 294 Introductiontothetheoryofoperatorspaces,G.PISIER 295 GeometryandIntegrability,L.MASON&YAVUZNUTKU(eds) 296 Lecturesoninvarianttheory,I.DOLGACHEV 297 Thehomotopycategoryofsimplyconnected4-manifolds,H.-J.BAUES 298 Higheroperads,highercategories,T.LEINSTER 299 KleinianGroupsandHyperbolic3-ManifoldsY.KOMORI,V.MARKOVIC&C.SERIES(eds) 300 IntroductiontoM¨obiusDifferentialGeometry,U.HERTRICH-JEROMIN 301 StableModulesandtheD(2)-Problem,F.E.A.JOHNSON 302 DiscreteandContinuousNonlinearSchr¨odingerSystems,M.J.ABLORWITZ,B.PRINARI& A.D.TRUBATCH 303 NumberTheoryandAlgebraicGeometry,M.REID&A.SKOROBOGATOV(eds) 304 GroupsStAndrews2001inOxfordVol. 1,C.M.CAMPBELL,E.F.ROBERTSON&G.C.SMITH(eds) 305 GroupsStAndrews2001inOxfordVol. 2,C.M.CAMPBELL,E.F.ROBERTSON&G.C.SMITH(eds) 306 Peyresqlecturesongeometricmechanicsandsymmetry,J.MONTALDI&T.RATIU(eds) 307 SurveysinCombinatorics2003,C.D.WENSLEY(ed.) 308 Topology,geometryandquantumfieldtheory,U.L.TILLMANN(ed) 309 CoringsandComodules,T.BRZEZINSKI&R.WISBAUER 310 TopicsinDynamicsandErgodicTheory,S.BEZUGLYI&S.KOLYADA(eds) 311 Groups: topological,combinatorialandarithmeticaspects,T.W.MU¨LLER(ed) 312 FoundationsofComputationalMathematics,Minneapolis2002,FELIPECUCKERetal(eds) 313 Transcendentalaspectsofalgebraiccycles,S.MU¨LLER-STACH&C.PETERS(eds) 314 Spectralgeneralizationsoflinegraphs,D.CVETKOVIC,P.ROWLINSON&S.SIMIC 315 Structuredringspectra,A.BAKER&B.RICHTER(eds) 316 LinearLogicinComputerScience,T.EHRHARDetal(eds) 317 Advancesinellipticcurvecryptography,I.F.BLAKE,G.SEROUSSI& N.SMART 318 Perturbationoftheboundaryinboundary-valueproblemsofPartialDifferentialEquations,D.HENRY 319 DoubleAffineHeckeAlgebras,I.CHEREDNIK 321 SurveysinModernMathematics,V.PRASOLOV&Y.ILYASHENKO(eds) 322 Recentperspectivesinrandommatrixtheoryandnumbertheory,F.MEZZADRI&N.C.SNAITH(eds) 323 Poissongeometry,deformationquantisationandgrouprepresentations,S.GUTTetal(eds) 324 SingularitiesandComputerAlgebra,C.LOSSEN&G.PFISTER(eds) 325 LecturesontheRicciFlow,P.TOPPING 326 ModularRepresentationsofFiniteGroupsofLieType,J.E.HUMPHREYS 328 FundamentalsofHyperbolicManifolds,R.D.CANARY,A.MARDEN&D.B.A.EPSTEIN(eds) 329 SpacesofKleinianGroups,Y.MINSKY,M.SAKUMA&C.SERIES(eds) 330 NoncommutativeLocalizationinAlgebraandTopology,A.RANICKI(ed) 331 FoundationsofComputationalMathematics,Santander2005,L.PARDO,A.PINKUS,E.SULI&M. TODD(eds) 332 HandbookofTiltingTheory,L.ANGELERIHU¨GEL,D.HAPPEL&H.KRAUSE(eds) 333 SyntheticDifferentialGeometry2ed,A.KOCK 334 TheNavier-StokesEquations,P.G.DRAZIN&N.RILEY 335 LecturesontheCombinatoricsofFreeProbability,A.NICA&R.SPEICHER 336 IntegralClosureofIdeals,Rings,andModules,I.SWANSON&C.HUNEKE 337 MethodsinBanachSpaceTheory,J.M.F.CASTILLO&W.B.JOHNSON(eds) 338 SurveysinGeometryandNumberTheoryN.YOUNG(ed) LondonMathematicalSocietyLectureNoteSeries:340 Groups St Andrews 2005 Volume 2 Edited by C.M. Campbell University of St Andrews M.R. Quick University of St Andrews E.F. Robertson University of St Andrews G.C. Smith University of Bath CAMBRIDGEUNIVERSITYPRESS Cambridge,NewYork,Melbourne,Madrid,CapeTown,Singapore,S˜aoPaulo CambridgeUniversityPress TheEdinburghBuilding,CambridgeCB22RU,UK PublishedintheUnitedStatesofAmericabyCambridgeUniversityPress,NewYork www.cambridge.org Informationonthistitle: www.cambridge.org/9780521694698 (cid:2)c CambridgeUniversityPress,2007 Thispublicationisincopyright. Subjecttostatutoryexception andtotheprovisionsofrelevantcollectivelicensingagreements, noreproductionofanypartmaytakeplacewithout thewrittenpermissionofCambridgeUniversityPress. PrintedintheUnitedKingdomattheUniversityPress,Cambridge AcataloguerecordforthispublicationisavailablefromtheBritishLibrary ISBN-13 978-0-521-69470-4paperback ISBN-10 0-521-69470-1paperback CambridgeUniversityPresshasnoresponsibilityforthepersistenceoraccuracyofURLsforexternalorthird- partyinternetwebsitesreferredtointhispublication,anddoesnotguaranteethatanycontentonsuchwebsites is,orwillremain,accurateorappropriate. Contents of Volume 2 Introduction page xi Groups and semigroups: connections and contrasts John Meakin 357 Toward the classification of s-arc transitive graphs A´kos Seress 401 Non-cancellation group computation for some finitely generated nilpotent groups Habtay Ghebrewold 415 Permutation and quasi-permutation representations of the Chevalley groups Maryam Ghorbany 424 The shape of solvable groups with odd order S. P. Glasby 432 Embedding in finitely presented lattice-ordered groups: explicit presentations for constructions A. M. W. Glass, Vincenzo Marra & Daniele Mundici 438 A note on abelian subgroups of p-groups George Glauberman 445 On kernel flatness Akbar Golchin 448 On proofs in finitely presented groups George Havas & Colin Ramsay 457 Computing with 4-Engel groups George Havas & M. R. Vaughan-Lee 475 On the size of the commutator subgroup in finite groups Marcel Herzog, Gil Kaplan & Arieh Lev 486 Groups of infinite matrices Waldemar Ho(cid:2)lubowski 491 v vi Contents Triply factorised groups and nearrings Peter Hubert 496 On the space of cyclic trigonal Riemann surfaces of genus 4 Milagros Izquierdo & Daniel Ying 504 On simple K -groups for n=5, 6 n A. Jafarzadeh & A. Iranmanesh 517 Products of Sylow subgroups and the solvable radical Gil Kaplan & Dan Levy 527 On commutators in groups Luise-Charlotte Kappe & Robert Fitzgerald Morse 531 Inequalities for the Baer invariant of finite groups Saeed Kayvanfar 559 Automorphisms with centralizers of small rank Evgeny Khukhro & Victor Mazurov 564 2-signalizers and normalizers of Sylow 2-subgroups in finite simple groups Anatoly S. Kondratiev & Victor D. Mazurov 586 On properties of abnormal and pronormal subgroups in some infinite groups Leonid A. Kurdachenko, Javier Otal & Igor Ya. Subbotin 597 P-localizing group extensions Karl Lorensen 605 On the n-covers of exceptional groups of Lie type Maria Silvia Lucido 621 Positively discriminating groups O. Macedon´ska 624 Automorphism groups of some chemical graphs G. A. Moghani, A. R. Ashrafi, S. Naghdi & M. R. Admadi 630 On c-normal subgroups of some classes of finite groups Z. Mostaghim 640 Fong characters and their fields of values Lucia Sanus 644 Arithmetical properties of finite groups W. J. Shi 646 On prefrattini subgroups of finite groups: a survey X. Soler-Escriva` 654 Frattini extensions and class field theory Th. Weigel 661 The nilpotency class of groups with fixed point free automorphisms of prime order Lawrence Wilson 685 Contents of Volume 1 Introduction page xi Aspects of infinite permutation groups Peter J. Cameron 1 Self-similarity and branching in group theory Rostislav Grigorchuk & Zoran Sˇuni´c 36 On surface groups: motivating examples in combinatorial group theory Peter Ackermann, Benjamin Fine & Gerhard Rosenberger 96 Nilpotent p-algebras and factorized p-groups Bernhard Amberg & Lev Kazarin 130 Classification of finite groups by the number of element centralizers Ali Reza Ashrafi & Bijan Taeri 148 Algorithmic use of the Mal’cev correspondence Bjo¨rn Assmann 158 Minimal but inefficient presentations for semi-direct products of finite cyclic monoids Firat Ate¸s & A. Sinan C¸evik 170 The modular isomorphism problem for finite p-groups with a cyclic subgroup of index p2 Czesl(cid:2)aw Bagin´ski & Alexander Konovalov 186 On one-generated formations A. Ballester-Bolinches, Clara Calvo & R. Esteban-Romero 194 New results on products of finite groups A. Ballester-Bolinches, John Cossey & M. C. Pedraza-Aguilera 199 Radical locally finite T-groups A. Ballester-Bolinches, H. Heineken & Tatiana Pedraza 203 Explicit tilting complexes for the Brou´e conjecture on 3-blocks Ayala Bar-Ilan, Tzviya Berrebi, Genadi Chereshnya, Ruth Leabovich, Mikhal Cohen & Mary Schaps 207 vii viii Contents Conjugacy classes of p-regular elements in p-solvable groups Antonio Beltr´an & Mar´ıa Jos´e Felipe 224 An algorithm for the unit group of the Burnside ring of a finite group Robert Boltje & Go¨tz Pfeiffer 230 Integral group ring of the first Mathieu simple group Victor Bovdi & Alexander Konovalov 237 Embedding properties in direct products B. Brewster, A. Mart´ınez-Pastor & M. D. P´erez-Ramos 246 Malcev presentations for subsemigroups of groups --- a survey Alan J. Cain 256 Finite groups with extremal conditions on sizes of conjugacy classes and on degrees of irreducible characters David Chillag & Marcel Herzog 269 Conjugacy class structure in simple algebraic groups Martin Cook 274 On automorphisms of products of groups Jill Dietz 288 Linear groups with infinite central dimension Martyn R. Dixon & Leonid A. Kurdachenko 306 G-automata, counter languages and the Chomsky hierarchy Murray Elder 313 An embedding theorem for groups universally equivalent to free nilpotent groups Benjamin Fine, Anthony M. Gaglione & Dennis Spellman 319 Irreducible word problems in groups Ana R. Fonseca, Duncan W. Parkes & Richard M. Thomas 327 Recent growth results Eric M. Freden & Teresa Knudson 341 Introduction Groups St Andrews 2005 was held in the University of St Andrews from 30 July to 6 August 2005. This was the seventh in the series of Groups St Andrews group theory conferences organised by Colin Campbell and Edmund Robertson of the UniversityofStAndrews. ThefirstthreewereheldinStAndrewsandsubsequent conferences held in Galway, Bath and Oxford before returning to St Andrews in 2005. We are pleased to say that the conference was, we believe, a success having been attended by 230 participants from 37 countries. The lectures and talks were given in the Mathematical Institute and the School of Physics and Astronomy of the University of St Andrews. Accommodation was provided in New Hall and in Fife Park. The Scientific Organising Committee of Groups St Andrews 2005 was: Colin Campbell (St Andrews), Nick Gilbert (Heriot-Watt), Steve Linton (St Andrews), John O’Connor (St Andrews), Edmund Robertson (St Andrews), Nik Ruskuc (St Andrews), Geoff Smith (Bath). The Committee received very valuable support from our Algebra colleagues at St Andrews, both staff and postgraduate students. Onceagain,webelievethatthesupportofthetwomainBritishmathematicssoci- eties, the Edinburgh Mathematical Society and the London Mathematical Society has been an important factor in the success of these conferences. ThemainspeakersatthemeetingwerePeterJCameron(QueenMary,London), RostislavIGrigorchuk(TexasA&M),JohnCMeakin(Nebraska-Lincoln)andAkos Seress (Ohio State). Additionally there were seven one-hour invited speakers to- gether with an extensive programme of over a hundred seminars; a lot to fit into a week! As has become the tradition, all the main speakers have written substan- tial articles for these Proceedings. Each volume begins with two such articles. All papers have been subjected to a formal refereeing process comparable to that of a major international journal. Publishing constraints have forced the editors to exclude some very worthwhile papers, and this is of course a matter of regret. It is hoped that the next conference in this series will be held in 2009 and will be Groups St Andrews in Bath. As Colin and Edmund will have retired before GroupsStAndrews2009,theylookforwardtothisconferencewithspecialinterest and anticipate a slightly different role. These Proceedings do, however, mark the first 25 years of Groups St Andrews conferences. In addition to the mathematics, illustrated by these Proceedings, the seven conferences have contained a wide se- lectionofsocialevents. ‘‘GroupsStAndrewstourism’’hastakenustoavarietyof ix