This page intentionally left blank LONDONMATHEMATICALSOCIETYLECTURENOTESERIES ManagingEditor:ProfessorM.Reid,MathematicsInstitute,UniversityofWarwick,CoventryCV47AL,UnitedKingdom Thetitlesbelowareavailablefrombooksellers,orfromCambridgeUniversityPressatwww.cambridge.org/mathematics 234 Introductiontosubfactors, V.JONES&V.S.SUNDER 235 Numbertheory:Se´minairedethe´oriedesnombresdeParis1993–94, S.DAVID(ed) 236 TheJamesforest, H.FETTER&B.GAMBOADEBUEN 237 Sievemethods,exponentialsums,andtheirapplicationsinnumbertheory, G.R.H.GREAVESetal(eds) 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.M.EVANS(ed) 245 Geometry,combinatorialdesignsandrelatedstructures, J.W.P.HIRSCHFELDetal(eds) 246 p-Automorphismsoffinitep-groups, E.I.KHUKHRO 247 Analyticnumbertheory, Y.MOTOHASHI(ed) 248 TametopologyandO-minimalstructures, L.VANDENDRIES 249 Theatlasoffinitegroups-Tenyearson, R.T.CURTIS&R.A.WILSON(eds) 250 Charactersandblocksoffinitegroups, G.NAVARRO 251 Gro¨bnerbasesandapplications, B.BUCHBERGER&F.WINKLER(eds) 252 Geometryandcohomologyingrouptheory, P.H.KROPHOLLER,G.A.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¨LKLEIN,J.G.THOMPSON,D.HARBATER&P.MU¨LLER(eds) 257 Anintroductiontononcommutativedifferentialgeometryanditsphysicalapplications(2ndedition), J.MADORE 258 Setsandproofs, S.B.COOPER&J.K.TRUSS(eds) 259 Modelsandcomputability, S.B.COOPER&J.TRUSS(eds) 260 GroupsStAndrews1997inBathI, C.M.CAMPBELLetal(eds) 261 GroupsStAndrews1997inBathII, C.M.CAMPBELLetal(eds) 262 Analysisandlogic, C.W.HENSON,J.IOVINO,A.S.KECHRIS&E.ODELL 263 Singularitytheory, W.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 Ergodictheoryandtopologicaldynamicsofgroupactionsonhomogeneousspaces, M.B.BEKKA&M.MAYER 271 Singularperturbationsofdifferentialoperators, S.ALBEVERIO&P.KURASOV 272 Charactertheoryfortheoddordertheorem, T.PETERFALVI.TranslatedbyR.SANDLING 273 Spectraltheoryandgeometry, E.B.DAVIES&Y.SAFAROV(eds) 274 TheMandelbrotset,themeandvariations, T.LEI(ed) 275 Descriptivesettheoryanddynamicalsystems, M.FOREMAN,A.S.KECHRIS,A.LOUVEAU&B.WEISS(eds) 276 Singularitiesofplanecurves, E.CASAS-ALVERO 277 Computationalandgeometricaspectsofmodernalgebra, M.ATKINSONetal(eds) 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.B.FU&R.W.OGDEN(eds) 284 Foundationsofcomputationalmathematics, R.DEVORE,A.ISERLES&E.SU¨LI(eds) 285 Rationalpointsoncurvesoverfinitefields, H.NIEDERREITER&C.XING 286 Cliffordalgebrasandspinors(2ndEdition), P.LOUNESTO 287 TopicsonRiemannsurfacesandFuchsiangroups, E.BUJALANCE,A.F.COSTA&E.MART´INEZ(eds) 288 Surveysincombinatorics,2001, J.W.P.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 Introductiontooperatorspacetheory, G.PISIER 295 Geometryandintegrability, L.MASON&Y.NUTKU(eds) 296 Lecturesoninvarianttheory, I.DOLGACHEV 297 Thehomotopycategoryofsimplyconnected4-manifolds, H.-J.BAUES 298 Higheroperads,highercategories, T.LEINSTER(ed) 299 Kleiniangroupsandhyperbolic3-manifolds, Y.KOMORI,V.MARKOVIC&C.SERIES(eds) 300 IntroductiontoMo¨biusdifferentialgeometry, U.HERTRICH-JEROMIN 301 StablemodulesandtheD(2)-problem, F.E.A.JOHNSON 302 DiscreteandcontinuousnonlinearSchro¨dingersystems, M.J.ABLOWITZ,B.PRINARI&A.D.TRUBATCH 303 Numbertheoryandalgebraicgeometry, M.REID&A.SKOROBOGATOV(eds) 304 GroupsStAndrews2001inOxfordI, C.M.CAMPBELL,E.F.ROBERTSON&G.C.SMITH(eds) 305 GroupsStAndrews2001inOxfordII, C.M.CAMPBELL,E.F.ROBERTSON&G.C.SMITH(eds) 306 Geometricmechanicsandsymmetry, 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, F.CUCKERetal(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.EHRHARD,P.RUET,J.-Y.GIRARD&P.SCOTT(eds) 317 Advancesinellipticcurvecryptography, I.F.BLAKE,G.SEROUSSI&N.P.SMART(eds) 318 Perturbationoftheboundaryinboundary-valueproblemsofpartialdifferentialequations, D.HENRY 319 DoubleaffineHeckealgebras, I.CHEREDNIK 320 L-functionsandGaloisrepresentations, D.BURNS,K.BUZZARD&J.NEKOVA´Rˇ(eds) 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 327 Surveysincombinatorics2005, B.S.WEBB(ed) 328 Fundamentalsofhyperbolicmanifolds, R.CANARY,D.EPSTEIN&A.MARDEN(eds) 329 SpacesofKleiniangroups, Y.MINSKY,M.SAKUMA&C.SERIES(eds) 330 Noncommutativelocalizationinalgebraandtopology, A.RANICKI(ed) 331 Foundationsofcomputationalmathematics,Santander2005, L.MPARDO,A.PINKUS,E.SU¨LI&M.J.TODD(eds) 332 Handbookoftiltingtheory, L.ANGELERIHU¨GEL,D.HAPPEL&H.KRAUSE(eds) 333 Syntheticdifferentialgeometry(2ndEdition), A.KOCK 334 TheNavier–Stokesequations, N.RILEY&P.DRAZIN 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 Surveysingeometryandnumbertheory, N.YOUNG(ed) 339 GroupsStAndrews2005I, C.M.CAMPBELL,M.R.QUICK,E.F.ROBERTSON&G.C.SMITH(eds) 340 GroupsStAndrews2005II, C.M.CAMPBELL,M.R.QUICK,E.F.ROBERTSON&G.C.SMITH(eds) 341 Ranksofellipticcurvesandrandommatrixtheory, J.B.CONREY,D.W.FARMER,F.MEZZADRI& N.C.SNAITH(eds) 342 Ellipticcohomology, H.R.MILLER&D.C.RAVENEL(eds) 343 AlgebraiccyclesandmotivesI, J.NAGEL&C.PETERS(eds) 344 AlgebraiccyclesandmotivesII, J.NAGEL&C.PETERS(eds) 345 Algebraicandanalyticgeometry, A.NEEMAN 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.MO¨RTERS&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.GARC´IA-PRADA& S.RAMANAN(eds) 360 Zariskigeometries, B.ZILBER 361 Words:Notesonverbalwidthingroups, D.SEGAL 362 Differentialtensoralgebrasandtheirmodulecategories, R.BAUTISTA,L.SALMERO´N&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.GONZA´LEZ-DIEZ&C.KOUROUNIOTIS(eds) 369 Epidemicsandrumoursincomplexnetworks, M.DRAIEF&L.MASSOULIE´ 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,regularity,andcompleteintersection, J.MAJADAS&A.G.RODICIO 374 Geometricanalysisofhyperbolicdifferentialequations:Anintroduction, S.ALINHAC 375 Triangulatedcategories, T.HOLM,P.JØRGENSEN&R.ROUQUIER(eds) 376 Permutationpatterns, S.LINTON,N.RUSˇKUC&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.KRAJ´ICˇEK 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) LondonMathematicalSocietyLectureNotesSeries:388 Groups St Andrews 2009 in Bath Volume 2 Editedby C. M. CAMPBELL UniversityofStAndrews M. R. QUICK UniversityofStAndrews E. F. ROBERTSON UniversityofStAndrews C. M. RONEY-DOUGAL UniversityofStAndrews G. C. SMITH UniversityofBath G. TRAUSTASON UniversityofBath cambridge university press Cambridge,NewYork,Melbourne,Madrid,CapeTown, Singapore,Sa˜oPaulo,Delhi,Tokyo,MexicoCity CambridgeUniversityPress TheEdinburghBuilding,CambridgeCB28RU,UK PublishedintheUnitedStatesofAmericabyCambridgeUniversityPress,NewYork www.cambridge.org Informationonthistitle:www.cambridge.org/9780521279048 (cid:1)C CambridgeUniversityPress2011 Thispublicationisincopyright.Subjecttostatutoryexception andtotheprovisionsofrelevantcollectivelicensingagreements, noreproductionofanypartmaytakeplacewithoutthewritten permissionofCambridgeUniversityPress. Firstpublished2011 PrintedintheUnitedKingdomattheUniversityPress,Cambridge AcataloguerecordforthispublicationisavailablefromtheBritishLibrary ISBN978-0-521-27904-8Paperback CambridgeUniversityPresshasnoresponsibilityforthepersistenceor accuracyofURLsforexternalorthird-partyinternetwebsitesreferredto inthispublication,anddoesnotguaranteethatanycontentonsuch websitesis,orwillremain,accurateorappropriate. CONTENTS Volume 2 Introduction...................................................................ix Algorithms for matrix groups E. A. O’Brien...............................................................297 Residual properties of 1-relator groups Mark Sapir ..................................................................324 Words and groups Dan Segal ...................................................................344 The modular isomorphism problem for the groups of order 512 Bettina Eick & Alexander Konovalov ........................................375 Recent progress in the symmetric generation of groups Ben Fairbairn................................................................384 Discriminating groups: a comprehensive overview Benjamin Fine, Anthony M. Gaglione, Alexei Myasnikov, Gerhard Rosenberger & Dennis Spellman.............................................................395 Extending the Kegel Wielandt theorem through π-decomposable groups L. S. Kazarin, A. Mart´ınez-Pastor & M. D. P´erez-Ramos....................415 On the prime graph of a finite group Behrooz Khosravi............................................................424 Applications of Lie rings with finite cyclic grading E. I. Khukhro................................................................429 Pronormal subgroups and transitivity of some subgroup properties Leonid A. Kurdachenko, Javier Otal & Igor Ya. Subbotin ....................448 On Engel and positive laws O. Macedon´ska & W. Tomaszewski ..........................................461 Maximal subgroups of odd index in finite groups with simple classical socle N. V. Maslova...............................................................473 Some classic and nearly classic problems on varieties of groups Peter M. Neumann ..........................................................479 Generalizations of the Sylow theorem Danila O. Revin & Evgeny P. Vdovin........................................488 Engel groups Gunnar Traustason..........................................................520 Lie methods in Engel groups Michael Vaughan-Lee ........................................................551 On the degree of commutativity of p-groups of maximal class A. Vera-L´opez & M. A. Garc´ıa-Sa´nchez......................................560 Class preserving automorphisms of finite p-groups: a survey Manoj K. Yadav.............................................................569 Symmetric colorings of finite groups Yuliya Zelenyuk..............................................................580 vi Contents of Volume 1 Introduction...................................................................ix A speech in honour of John Cannon and Derek Holt Charles Leedham-Green........................................................xi Finite groups of Lie type and their representations Gerhard Hiss...................................................................1 Iterated monodromy groups Volodymyr Nekrashevych......................................................41 Engel elements in groups Alireza Abdollahi..............................................................94 Some classes of finite semigroups with kite-like egg-boxes of D-classes K. Ahmadidelir & H. Doostie................................................118 Structure of finite groups having few conjugacy class sizes Antonio Beltr´an & Mar´ıa Jos´e Felipe ........................................124 Group theory in cryptography Simon R. Blackburn, Carlos Cid & Ciaran Mullan...........................133 A survey of recent results in groups and orderings: word problems, embeddings and amalgamations V. V. Bludov & A. M. W. Glass.............................................150 A survey on the minimum genus and maximum order problems for bordered Klein surfaces E. Bujalance, F. J. Cirre, J. J. Etayo, G. Gromadzki & E. Mart´ınez.........161 On one-relator quotients of the modular group Marston Conder, George Havas & M. F. Newman............................183 Miscellaneous results on supersolvable groups K. Corra´di, P. Z. Hermann, L. H´ethelyi & E. Horva´th.......................198 Automorphisms of products of finite groups M. John Curran.............................................................213 vii A rational property of the irreducible characters of a finite group M. R. Darafsheh, A. Iranmanesh & S. A. Moosavi...........................224 Automotives Marian Deaconescu & Gary Walls...........................................228 On n-abelian groups and their generalizations Costantino Delizia & Antonio Tortora .......................................244 Computing with matrix groups over infinite fields A. S. Detinko, B. Eick & D. L. Flannery ....................................256 Trends in infinite dimensional linear groups Martyn R. Dixon, Leonid A. Kurdachenko, Jose M. Mun˜oz-Escolano & Javier Otal .........................................................................271 Engel conditions on orderable groups and in combinatorial problems (a survey) Marcel Herzog, Patrizia Longobardi & Mercede Maj ..........................283 viii