cxdrespectivelyItremainedanopenproblemwhetherthesameholdsforin nitesharply2and3transitivegroupsandmuchliteratureonthistopicisavailableseeRSTforbackgroundandmorerecentreferencesInRSTthe rstc ID: 427151
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Sharply3-transitivegroupsKatrinTentAugust6,2015AbstractWeconstructtherstsharply3-transitivegroupsnotarisingfromaneareld,i.e.pointstabilizershavenonontrivialabeliannormalsubgroup.11IntroductionThenitesharply2-and3-transitivegroupswereclassiedbyZassenhausin[Z1]and[Z2]inthe1930'sandwereshowntoarisefromso-callednear-elds.Theyessentiallylooklikethegroupsofanelineartransformationsx7!ax+borMoebiustransformationsx7!ax+b cx+d,respectively.Itremainedanopenproblemwhetherthesameholdsforinnitesharply2-and3-transitivegroupsandmuchliteratureonthistopicisavailable,see[RST]forbackgroundandmorerecentreferences.In[RST]therstconstructionofsharply2-transitivegroupswithoutanynontrivialabeliannormalsubgroupisgiven.Howeverthequestionremainedopenwhetherthegroupsconstructedtherecanbeextendedtogroupsactingsharply3-transitively.Wehereuseadi erentapproachdirectlyconstructingsharply3-transitivegroupsusingpartialgroupactions.Thesearetherstknownexamplesofsharply3-transitivegroupswhosepointstabilizershavenonon-trivialabeliannormalsubgroupandthusdonotarisefromnear-elds.ByresultsofTits[Ti]andHall[Ha]therearenoinnitesharplyk-transitivegroupsfork4,seee.g.[DM]. PartiallysupportedbySFB8781Keywords:permutationgroups,sharply3-transitiveactions,amalgamatedproducts,neareld1 2ThemaintheoremForbrevitywecallanelementoforder3a3-cycleandwesaythatagroupactionis3-sharpifall3-pointstabilizersaretrivial.Theorem2.1.LetG0beagroupinwhichall3-cyclesandallinvolutions,respectively,areconjugateandsuchthatthereexistsa3-cycleaandanin-volutiontinG0withha;ti=S3.AssumethatG0actsonasetX0insuchawaythat1.theactionis3-sharp;2.theinvolutiont2G0xesauniquepointx02X0;3.the3-cyclea2G0isxedpointfree;4.wehave(x0;x0a;x0a2)=(x0;x0a;x0at);5.ifB=(x;y;z)isatripleinX0suchthatthesetwisestabilizerofBinG0isisomorphictoS3,thenthereissomeg2G0withAg=BwhereA=(x0;x0a;x0a2)=(x0;x0a;x0at).ThenwecanextendG0toasharply3-transitiveactionofG=�(haiF(U))haiG0hti((htiF(S)))F(R)onasuitablesetYX0,whereF(R);F(S);F(U)arefreegroupsondisjointsetsR;S;UwithjRj;jSj;jUj=maxfjG0j;@0g.Remark2.2.NotethatS3initsnaturalactiononthreeelementssatisestheassumptionsofTheorem2.1(asdoesPGL(2;2)=S3actingasasubgroupofPGL(2;23)ontheprojectivelineoverF23.)ThereforewehaveCorollary2.3.ThereexistgroupsGactingsharply3-transitivelyonsomesetXsuchthatforx2XthepointstabilizerGxofxhasnonontrivialabeliannormalsubgroup.Proof.ApplyingTheorem2.1toG0=S3=haiohtiweseethatnotwodistinctinvolutionsinGcommute.ThepointstabilizerGxactssharply2-transitivelyonXnfxg.SincetheinvolutionsinG0andhence(byconjugacy)2 alsoinGhaveuniquexedpoints,theinvolutionsofGxintheiractiononXnfxgarexedpointfree,sothesharply2-transitivegroupGxissaidtohavecharacteristic2.By[Ne](seealsoe.g.[BN],11.46)anontrivialabeliannormalsubgroupofGxwouldhavetoconsistofelementsoforder2,eachbeingtheproductoftwoinvolutionsinGx.FromtheconstructionofG,itcanbeseenthatthisisimpossibleinG. 3TheconstructionInthissectionweproveTheorem2.1,i.e.weconstructGanditsactiononasetXfromG0andX0.Wecontinuetousethenotationintroducedintheprevioussection.Fortheconstructionweusepartialgroupactionsasin[TZ],consideredalsobyRipsandSegev.TheconstructionproceedsbyextendinginductivelyboththegroupG0andtheunderlyingsetX0insuchawaythattheassump-tionsofTheorem2.1arepreserved.Thisisdoneintwoseparatekindsofextensions:ontheonehandweextendthegroupG0inordertomaketheactionabitmore3-transitive.OntheotherhandweextendtheunderlyingsetX0inordertolettheextendedgroupact,forcingusinturntoextendthegroupagainetc.Inthelimit,thegroupGwillbesharply3-transitiveonasetXcontainingX0.Denition3.1.ApartialactionofGonasetXcontainingX0consistsofanactionofG0onXandpartialactionsofthegeneratorsinS[R[Usuchthat1.fors2Swehavex0s=x0andifxsisdenedforx2Xnfx0g,thensois(xt)sandwehave(xt)s=(xs)t.2.foru2U;x2X,ifxuisdened,thensoare(xa)uand(xa2)uandwehave(xa)u=(xu)aand(xa2)u=(xu)a2.Wenowdenenormalformssuitableforourpurpose:Normalforms3.2.AnyelementofGcanbewritten(notnecessarilyuniquely)asareducedwordinthegeneratorsG0nf1g[S[S�1[R[R�1[U[U�1wherewesaythatawordisreducediftherearenosubwordsoftheformff�1forf2S[S�1[R[R�1[U[U�1,3 ghforg;h2G0nf1g;s1ts2wheres1;s22S[S�1;ts1sng(oritsinverse)wheresi2S[S�1;i=1;:::;n;g2G0nf1g;u1a1u2whereu1;u22U[U�1;a1u1ungoritsinversewhereui2U[U�1;i=1;:::;n;g2G0nf1g.Thewordwiscalledcyclicallyreducedifwandeverycyclicpermutationofwisreduced.Wesaythatforawordw=s1sninG0nf1g[S[S�1[R[R�1[U[U�1,theelementxwisdenedforx2Xifforallinitialsegmentsofwtheactiononxisdened,i.e.xs1;:::;(:::(xs1):::)si;in,aredenedandwewritexw=(:::(xs1):::)sn.NoticethatforelementsfromG0theactiononXisdenedeverywhere.Henceifw;w0arereducedwordswithw=w0inGandxwisdened,thenxw0isdenedaswellandxw=xw0.Thustheexpressionxg=ymakessenseforg2G;x;y2X.IfGactspartiallyonX,thenthereisacanonicalpartialactiononthesetoftriples(X)3=f(x;y;z)2X3jjfx;y;zgj=3g:TerminologyandnotationForatripleC=(x;y;z)wesaythatg2GshiftsCifCg=(y;z;x)orCg=(z;x;y)andwesaythath2G ipsCifCh=(x;z;y)orashiftofthis.Notethatiftherearesuchelementsg;h2GthenthesetwisestabilizerofCinGisisomorphictoS3.Wealsosaythatanelementg2GleavesatripleCinvariantifCgisdenedandisequaltoCasaset.InthiscasewecallCag-triple.Wecallatriplefreeiftheonlyelementleavingitinvariantistheidentity.WecalltwotriplesCandC0connectedifthereisw2GsuchthatCwisdenedandequalsC0.Ifw2GleavesatripleCinvariant,thenwritingw=s1:::sninreducedform,thetriples(C;Cs1;Cs1s2;:::;Cw)formacyclewhichwecallbraidedifCwisequaltoCasaset,butnotnecessarilyasanorderedtriple.Denition3.3.WecallapartialactionofGonXgood(andsaythatGactswellonX)if4 1.the3-cycleaactswithoutxedpointonX;2.theinvolutionthasauniquexedpoint,namelyx02X;andforalltriplesC2(X)3andg;h2Gthefollowingholds:3.Cg=Cimpliesg=1.4.Ifh ipsC,thenhisaconjugateoft.5.IfgshiftsC,thengisaconjugateofa.6.IfgshiftsCandh ipsC,thenthereissomeg02GsuchthatCg0=AwhereA=(x0;x0a;x0a2)=(x0;x0a;x0at).NotethattheoriginalactionofG0onX=X0isgoodandthatAistheuniquetripleofX0invariantunderbothaandt.Inordertomaketheaction3-transitiveitsucestoconnectAtoanyothertripleoftheunderlyingset.Noticethatsinceadoesnotxanypoint,wehave(x;xa;xa2)2(X)3forallx2X.Intheremainderofthesectionweextendagoodpartialactionintwoways:bylettingfreegeneratorstakethe(distinguished)tripleAtoalla-,t-andfreetriplesinordertoeventuallymaketheaction3-transitive,andbyextendingthedomainofthepartialactionofafreegeneratorinordertoeventuallymaketheactiontotal.Westartwiththelastone:Lemma3.4(Extendingthefreegenerators).AssumethatGactswellonXandthatforsomex2Xandf2S[S�1[U[U�1[R[R�1theexpressionxfisnotdened.(Thenxtfisnotdenediff2S[S�1andxaf;xa2farenotdenedincasef2U[U�1.)Letx0G0=fx0g0jg02G0gbeasetofnewelementsonwhichG0actsregularlyandextendthepartialoperationofGtoX0=X[x0G0byputting1.xf=x0iff2R[R�1;2.xf=x0and(xt)f=x0t(andx0f=x0)iff2S[S�1;3.xf=x0,(xa)f=x0aandxa2f=x0a2iff2U[U�1.ThenGactswellonX0.5 Proof.FirstobservethatthisclearlydenesapartialactioninthesenseofDenition3.1andthattheactionsofaandtstillsatisfyconditions1.and2.ofDenition3.3.Fortheremainingconditionsof3.3itsucestoprovethatifacyclicallyreducedwordwleavesatripleCinX0invariant,thenw2G0(andhencewisconjugatetoaortbyassumption)orCandthe(possiblybraided)cycledescribedbywappliedtoCarecontainedinX.Sincethepreviousactionwasgood,thisisenough.Supposeotherwise:letw2GnG0becyclicallyreduced(inthesenseof3.2)leavingthetripleCinvariant.AssumethatatleastonetripleofthecyclegivenbyapplyingwtoCdoesnotbelongtoX.FirstassumethatthereisatripleinthecyclewhichdoesnotbelongtoX,butbothitsneighboursdo.Thiseasilyimpliesthatacyclicpermutationofwcontainsthesubwordff�1asfistheonlyelementtakingatriplefromXtoatriplenotentirelybelongingtoX.Sowisnotcyclicallyreduced,acontradiction.NextassumethattherearetwoneighbouringtriplesC1;C2inthecyclewhichdonotbelongtoX.ThenbythepropertiesofacyclicallyreducedwordandthedenitionofX0,C1andC2areconnectedbyanelementg2G0n1.Sothecyclecontainsasegment(B;C1;C2;D)wherethetriplesB;DarecontainedinXandnecessarilyBf=C1;C1g=C2andC2f�1=D.Thenf=2R[R�1asG0actsregularlyonx0G0.Iff2S[S�1,wemusthaveg=tsinceonx0G0theelementf�1isonlydenedonx0t.Soacyclicpermutationofwcontainsthesubwordftf�1,acontradiction.Similarly,iff2U[U�1,wehaveg=a1andacyclicpermutationofwcontainsthesubwordfa1f�1,againacontradiction.ThisshowsthatifatripleCbecomesinvariantundersomeg2Gundertheextendedaction,thisisinducedbyconjugationunderthepreviousaction.HenceCondition4.of3.3ispreservedaswell. WenextshowhowtoextendthegroupactioninordertoconnecttheuniquetripleA=(x0;x0a;x0a2)withx0at=x0a2toatripleBwhereBiseitherana-triple,at-tripleorafreetriple:Lemma3.5(ConnectingAtoothertriples).AssumethatGactswellonX,letA=(x0;x0a;x0a2)=(x0;x0a;x0at)beasbeforeandletBbeana-,t-orafreetripleforwhichthereisnog2GwithAg=B.Letf2R[S[Ubeanelementwhichdoesnotyetactanywherewith6 1.f2UifBisana-triple;2.f2SifBisat-triple;3.f2RifBisafreetripleExtendtheactionbysettingAf=B.ThenthisactionofGonXisagaingood.Proof.AgainitisclearthatthisdenesapartialactioninthesenseofDenition3.1andthat1.and2.ofDenition3.3continuetohold.AlsonotethatsincethesetwisestabilizerofAinGisS3theassumptionsimplythatthereisnow2Gnotcontainingf;f�1takingAtoBasaset.Toprovetheremainingconditionsof3.3letwbeacyclicallyreducedword(inthesenseof3.2)leavingsometripleC2(X)3invariant.Sincethepreviousactionwasgood,itsucestoshowthatwdoesnotcontainforf�1.Supposeotherwise.Thenbycyclicallypermutingwandtakinginverseswemayassumethatw=fw0.SowstabilizesAandw0takesBtoAasaset.ByassumptiononA;Bthesubwordw0mustcontainf.Hencewemaywritew0=v0fvforsomesubwordv0notcontainingforf�1.Wedistinguishtwocases:1.=1.Thenv0takesBtoAasasetasfisonlydenedonA.Sincev0doesnotcontainf;f�1,thiscontradictstheassumptiononA;B.2.=�1.Thenv0leavesBinvariant.Sincethepreviousactionwasgood,weeitherhavev0=tandf2Sorv0=a1andf2U(accordingtowhetherBisana-orat-triple).Ineithercasev0commuteswithf,contradictingtheassumptionthatwbecyclicallyreduced. Corollary3.6.AssumethatGactswellonXwithjXjmaxf@0;jGjgandtherearesucientlymanyelementsofR;SandUwhoseactionisnotyetdenedanywhere.ThenwecanextendthepartialactionofGonXtoasharply3-transitive(total)actiononsomeappropriatesupersetY.Proof.Fixtheuniquea-tripleA=(x0;x0a;x0a2)=(x0;x0a;x0at)inX0.UsingthepreviouslemmaswedeneasetYanda3-sharpactionofGonYwiththefollowingproperties:1.alla-triplesareconnectedtoA;7 2.allt-triplesareconnectedtoA;3.anytriple(x;y;z)canbeshiftedbyanelementofG.ThelastpropertycanbeachievedusingLemma3.5:supposeB=(x;y;z)cannotbeshiftedbyanelementofGatacertainstageoftheconstruction.ThenBisafreetriple:otherwiseanelementofGleavingBinvariantwouldhavetobeaninvolutiont02G.Sincet0=hth�1forsomeh2G,thetripleBhisat-tripleandhenceconnectedtoA,makingBshiftableaswell.Thus,Bisafreetripleatthatstage,andwelaterextendtheactionbyputtingAr=Bforsomer2R.ThenBcanbeshiftedbyar.ThiseasilyimpliesthattheactionofGonYissharply3-transitive,i.e.alltriplesareconnectedtoA:letBbeatripleandg2GshiftB.Thenwehaveg=hah�1forsomeh2G,soBhisana-tripleandwhenceconnectedtoA. ThisconcludestheproofofTheorem2.1.References[BN]A.Borovik,A.Nesin,GroupsofniteMorleyrank,OxfordLogicGuides,26.OxfordSciencePublications.TheClarendonPress,OxfordUniversityPress,NewYork,1994.[DM]J.D.Dixon,B.Mortimer,Permutationgroups,GraduateTextsinMathematics,163.Springer-Verlag,NewYork,1996.xii+346pp.[Ha]M.Hall,OnatheoremofJordan,PacicJ.ofMath.4(1954)219{226.[Ne]B.H.Neumann,OnthecommutativityofadditionJ.LondonMathSoc.15(1940),203{208.[RST]E.Rips,Y.Segev,K.Tent,Asharply2-transitivegroupwithoutanon-trivialabeliannormalsubgroup,toappearinJournaloftheEMS.[TZ]K.Tent,M.Ziegler,Sharply2-transitivegroups,preprint,2014.[Ti]J.Tits,Groupestriplementtransitifsetgeneralisations.AlgebreetThe-oriedenombres,Coll.Int.duCentreNat.delaRech.Sci.no.24(1950)207{208.8 [Z1]H.Zassenhaus,UberendlicheFastkorper,Abh.Math.Sem.Hamburg,11(1936),187{220.[Z2]H.Zassenhaus,KennzeichnungendlicherlinearerGruppenalsPermu-tationsgruppen,Abh.Math.Sem.Hamburg,11(1936),17{40.KatrinTent,MathematischesInstitut,UniversitatMunster,Einsteinstrasse62,D-48149Munster,Germany,tent@wwu.de9