Math.Log.Quart.51,No.6,626 - PDF document

Math.Log.Quart.51,No.6,626–631(2005)/DOI10.1002/malq.200510012/www.mlq-journal.orgjTj+-resplendentmodelsandtheLascargroupEnriqueCasanovas¤andRodrigoPel´aez¤¤DepartamentodeL´ogica,HistoriayFilosof´adelaCiencia,UniversidaddeBarcelona,BaldiriReixacs/n,E-08028Barcelona,SpainReceived14January2005,revised24August2005,accepted24August2005Publishedonline1October2005KeywordsAutomorphismgroups,Lascargroup,resplendentmodels.MSC(2000)03C45InthispaperweshowthatineveryjTj+-resplendentmodelN,foreveryAµNsuchthatjAj·jTj,thegroupAutf(N=A)ofstrongautomorphismsistheleastverynormalsubgroupofthegroupAut(N=A)andthequotientAut(N=A)=Autf(N=A)istheLascargroupoverA.ThenwegeneralizethisresulttoeveryjTj+-saturatedandstronglyjTj+-homogeneousmodel.c°2005WILEY-VCHVerlagGmbH&Co.KGaA,Weinheim1IntroductionLascarin[4]introducedthegroupAutf(N=A)ofstrongautomorphismsoverA,anormalsubgroupofthegroupAut(N=A)ofallautomorphismsoverAofamodelNcontainingA.ThequotientAut(N=A)=Autf(N=A)isindependentofthechoiceofN(forabigsaturatedmodelNandasmallsubsetAµN)anditisnowcal-ledtheLascargroupoverA.Lascarshowedin[4]thatinthecaseofaverylargeclassoftheories,calledbyhimG-compact,thegroupcarriesacompactHausdorfftopology.RecentlytheLascargrouphasreceivedalotofattention,particularlybecauseofitsimportanceforsimpletheoriesandhyperimaginariesandbecauseofthediscoveryofnon-G-compacttheories.Itisacompact(notnecessarilyHausdorff)topologicalgroupforanyrst-ordertheory,eveninanonG-compactone.InthecaseofthetheoryofanalgebraicallyclosedelditcorrespondstotheabsoluteGaloisgroupovertheeldgeneratedbyA,anditisapronitegroup.Wereferto[5,3,8]formoredetails.Thepresentationofthegroupin[4]isdoneintheframeworkofanuncountablesaturatedmodelNofanarbi-trarycountablecompleterst-ordertheoryandanitesubsetAµN.Itisstraightforwardtogeneralizeittoanycompleterst-ordertheoryTandanysaturatedmodelNofTwithjNj�jTj.ThusitalwayscanbeconstructedworkinginthemonstermodelCofT.Althoughthedetailshavenotbeenwritten,itisgenerallyacknowledgedthatinsteadofsaturatedmodelsonecanusespecialmodelsoftherightcardinality.Forinstance,Zieglerobservesin[8]thataspecialmodelNsuchthatcf(jNj)�2jTjissufcient.Theinconvenienceofworkingwithsaturatedmodelsisthatforsometheoriesitsexistencecannotbeprovenwithoutextrasettheoreticalhypotheses.Ontheotherhandspecialmodelsdoalwaysexist.WehavenoticedthatthereisamoregeneralclassofmodelswheretheLascargroupnaturallyarises:theclassofjTj+-resplendentmodels.Moreoverthepropertiesofthegroupofstrongautomorphismscanbeunderstoodmoreeasilyworkingwiththesemodels.ThenotionofresplendencyhasbeenintroducedbyBarwiseandSchlipfin[1].Poizatin[6]denedandstudiedthemoregeneralnotionof·-resplendency.InSection2wesummarizethemainfacts.jTj+-resplendentmodelsgeneralize(intherightcardinality)saturatedandspecialmodels,andinthecaseofstabletheoriestheycoincidewithsaturatedmodels.InunstabletheoriestherearemanyjTj+-resplendentmodelswhichareneithersaturatednorspecial.¤Correspondingauthor:e-mail:e.casanovas@ub.edu¤¤e-mail:rodrigopelaez@gmail.comc°2005WILEY-VCHVerlagGmbH&Co.KGaA,Weinheim Math.Log.Quart.51,No.6(2005)/www.mlq-journal.org627Wefocusonthepureabstractgroupsincethereisnothingnewconcerningthetopology.ThetopologyoftheLascargroupcanbeexplainedasin[8]usingonlythejTj+-saturationofthemodelandthepresenceofthepuregroup.InSection3westateandprovethemainresults.Theorem3.11showsthatanyjTj+-resplendentmodelNgivesrisetotheLascargroupandTheorem3.8indicatesthatthegroupofstrongautomorphismsoverAcanbecharacterized(similarlytowhatLascaroriginallydidworkingwithsaturatedmodels)astheleastverynormalsubgroupofAut(N=A),thatis,itsleastnormalsubgroupclosedunderamoregeneralconjugationthatwecallweakconjugation.Itshouldbenoticedthatthemethodsusedintheproofsarequitedifferent.Inparticularweneveruseultraproducts.FinallyinSection4weshowthattheseresultsalsoholdinthewiderclassofalljTj+-saturatedandstronglyjTj+-homogeneousmodels.Wewilldealwithanarbitrarycompleterst-ordertheoryTwithinnitemodels.ItslanguagewillbeLandCwillbeitsmonstermodel,whichwethinkofasmodelwhoseuniverseisaproperclassandwhichrealizesanytypeoveranysubset.Theexistenceofthemonstermodelcanbeguaranteedinanytheoryanddoesnotrequireanyadditionalhypothesis.AllmodelsweconsiderwillbeelementarysubmodelsofC.IfAµMisasetofparameters,L(A)istheexpandedlanguagewithnamesforallelementsofA,MAisthestandardexpansionofMtoL(A),whereeveryelementofAhasitscorrespondingnameandT(A)=Th(MA)isitsrst-ordertheory.AnA-automorphismofamodelM¶AisanautomorphismfofMwhichistheidentityonA.ItisalsocalledanautomorphismofMoverA.ThegroupofallA-automorphismsofMisAut(M=A).Whenwespeakofthetypetp(M=A)ofamodelMoverasetA,weimplicitlyassumeanenumerationofthemodelM.Asmentionedbefore,LascarworkedovernitesetsofparametersA.IfNissaturatedandjAjjNj,NAisagainsaturated.Thereforeonecanalsoworkoversuchsmallsubsets.ThesameappliestospecialmodelsNifcf(jAj)jNj.HerewewillconsidersubsetsAµNofjTj+-resplendentmodelsNwiththecardinalityrestrictionthatjAj·jTj.AgainNAisjTj+-resplendentinsuchasituation.InallthesecasesonecouldhaveassumedAtobeemptyworkinginT(A).Inourcasethiswouldhavesimpliedthenotationinthestatementsandproofs.HoweverwethinkitisimportanttopresentthetheoryoverasetAsinceitismainlyintendedtodealwiththiscaseandbecausesomedelicatequestionsconcerningpreservationofpropertiesoftheLascargroupwhenAchangesarestillopen.2jTj+-resplendencyDenition2.1Let·beaninnitecardinalnumberandletMbeanL-structure.WesaythatMis·-ex-pandableifforeverylanguageL0¶LwithjL0nLj·,if§isasetofsentencesoflanguageL0consistentwithTh(M),thenthereisanL0-expansionofMsatisfying§.WesaythatMis·-resplendentifandonlyifforeveryAµMwithjAj·,MAis·-expandable.Hereourinterestin·-resplendencyresidesinthecase·=jTj+.Hencewewillrestrictourattentiontothisparticularcaseinthenextresults,someofwhichcanbeeasilygeneralizedtoothercardinalnumbers·.Proposition2.21.ForeveryMthereissomejTj+-resplendentmodelNºMsuchthatjNj·jMj+2jTj.2.EverysaturatedmodelMsuchthatjMj�jTjandeveryspecialmodelMsuchthatcf(jMj)�jTjisjTj+-resplendent.3.jTj+-resplendentmodelsarestronglyjTj+-homogeneousandjTj+-saturated.4.IfTisstable,theneveryjTj+-resplendentmodelissaturated.5.IfTisunstable,thenforeveryMthereissomejTj+-resplendentmodelNºMofcardinalityjNj·jMj+2jTjwhichisnotjTj++-saturated.Proof.1.See[6,proofofTh´eoreme9.15].2.Shelahprovesin[7,ConclusionI.1.13]thateverysaturatedmodelofcardinality�jTjisjTj+-expandable.FromthisitfollowsimmediatelythatitisalsojTj+-resplendent.ThesamefactisalsoprovenbyPoizatin[6,Th´eoreme9.17].Thecaseofaspecialmodelisconsideredbytherstauthorin[2],whereinPropo-sition1.2itisestablishedthateveryspecialmodelofcardinality�jTjisjTj+-expandable.Now,ifMisspecialc°2005WILEY-VCHVerlagGmbH&Co.KGaA,Weinheim 628E.CasanovasandR.Pel´aez:jTj+-resplendentmodelsandtheLascargroupandAµMisofcardinalitycf(jMj),thenMAisstillspecialandhencejT(A)j+-expandable.ThereforeMisjTj+-expandableifcf(jMj)�jTj.3.ItisobviousthatajTj+-resplendentmodelMisjTj+-saturated.TocheckthatitisstronglyjTj+-homo-geneous,weconsiderapartialelementarymappingf,jfj·jTj,withdomainandrangecontainedinM,andweshowthatfcanbeextendedtoanautomorphismofM.LetA=dom(f),addanewunaryfunctionalsymbolF,putL0=L[fFgandnotethattheset§ofsentencesofL0(A[f(A)),expressingthata)Fisanautomorphism,b)F(a)=f(a)foralla2A,isconsistentwithT(A)=Th(MA).ByjTj+-resplendencythereisanexpansion(M;FM)ofMsuchthat(MA[f(A);FM)²§.Clearly,FMisanautomorphismofMextendingf.4.and5.areduetoPoizat.Theyfollowfrom[6,Th´eoreme16.11]and[6,Th´eoreme14.10]respectively.TheboundonthecardinalityofNin5.canbeobtainedfromtheproofgiventhere.Corollary2.3IfTisunstable,thenforeveryM,foreverycardinal·¸jMj+2jTjsuchthatcf(·)�jTj+,thereissomenonspecialmodelNºMofcardinality·whichisjTj+-resplendent.Proof.ByProposition2.2,5.,thereissomejTj+-resplendentmodelNºMofcardinality·whichisnotjTj++-saturated.Sincecf(·)�jTj+,everyspecialmodelofcardinality·isjTj++-saturated.HenceNisnotspecial.3TheLascargroupDenition3.1LetAµMandf;g2Aut(M=A).Wesaythatf;gareA-conjugateandwewritef»AgiftheyareconjugateelementsofthegroupAut(M=A),thatis,ifg="±f±"¡1forsome"2Aut(M=A).Wesaythatf;gareweaklyA-conjugateandwewritef¼AgifforsomeNºMthereareextensionsfµf02Aut(N=A)andgµg02Aut(N=A)suchthatf0»Ag0.WesaythatasubgroupGofAut(M=A)isverynormalifitisclosedunderweaklyconjugation,thatisifforanyf2Gandanyg2Aut(M=A)suchthatf¼Agwehaveg2G.Weuse[f]¼Atodenotethe¼A-classoff2Aut(M=A),i.e.[f]¼A=fg2Aut(M=A):f¼Agg:Thisnotationshouldnotsuggestthatweakconjugationisanequivalencerelation.Remark3.2Letf;g2Aut(M=A).Thenf¼Agifandonlyifthereareextensionsfµ¹f2Aut(C=A)andgµ¹g2Aut(C=A)suchthat¹f»A¹g.Verynormalsubgroupsarenormal.IngeneralfidMgisnotaverynormalsubgroupofAut(M=A),asshowninProposition3.5,butthereisasmallestverynormalsubgroup,beacusetheintersectionofanyfamilyofverynormalsubgroupsofAut(M=A)isagainaverynormalsubgroup.Denition3.3Wedenoteby¡(M=A)theintersectionofallverynormalsubgroupsofAut(M=A),whichisagainverynormal.Notethat¡(M=A)isaunionof¼A-classesandcontains[idM]¼A.Proposition3.4ForAµM0¹M,Aut(M=M0)µ[idM]¼A.Proof.Letf2Aut(M=M0)andexpandthelanguageL(M)byaddingthreenewunaryfunctionsym-bolsE;F;G.Let§bethesetofsentencesintheexpandedlanguageexpressingthat1.F;G;EareA-automorphisms;2.G=E¡1±F±E;3.G(m)=mforallm2M;4.f(m)=F(m)forallm2M.If§isconsistent,thenwehavenishedsincethereisanexpansionC0ofCMwhichsatises§andthenfµFC0»AGC0¶idM;whichshowsthatf¼AidM.Toshowtheconsistencyof§(withT(M))leta2Mbeanitetupleandletusc°2005WILEY-VCHVerlagGmbH&Co.KGaA,Weinheim Math.Log.Quart.51,No.6(2005)/www.mlq-journal.org629provethatthereare¹f;¹g;"2Aut(C=A)suchthat¹g="¡1±¹f±",¹g(a)=aand¹f(a)=f(a).Todothiswerstcheckthatifp(x)=tp(a=A),then(¤)p(x)[“xa´Axf(a)”isconsistent:Let'(x)2p(x).SinceAµM0,thereisb2M0suchthat²'(b).SincefistheidentityinM0,f(b)=bandhenceba´Abf(a).Thisensurestheconsistency.By(¤),thereisatuplea0´Aasuchthata0a´Aa0f(a).Nowchooseautomorphisms¹f;"2Aut(C=A)with"(a)=a0and¹f(a0a)=a0f(a).If¹g="¡1±¹f±",itfollowsthat¹g(a)=a.Proposition3.5IfNisjTj+-resplendentandAµNisofcardinalityatmostjTj,thenj[idN]¼Aj¸jNj.InparticularfidNgisnotaverynormalsubgroupofAut(N=A).Proof.ChooseanelementarysubmodelM¹NcontainingAofcardinality·jTjandchooseanonalge-braictypep(x)2S(M).InthemonstermodelwemayndaproperclassPofrealizationsofp(x).LetFbeanewbinaryfunctionalsymbol,letL0=L[fFg,andlookatthefollowingset§ofsentencesofL0(M):1.Forallx,themappingy7¡!F(x;y)isanautomorphism.2.8x(F(x;a)=a)foralla2M.3.8xy(x6=y!9z(F(x;z)6=F(y;z))).TheconsistencyofthissetofsentenceswithTh(NM)followsfromthefactthatinthemonstermodelwemayndforeachtwodifferenta;b2Panautomorphismf2Aut(C=M)withf(a)=b.Now,byjTj+-resplendencyofNthereisanexpansion(N;FN)ofNsuchthat(NM;FN)²§.Foreacha2Nwegetadifferentautomor-phismfa2Aut(N=M)denedbyfa(b)=FN(a;b).ByProposition3.4fa¼AidNforalla2N.Denition3.6LetAµM.ThegroupofallstrongautomorphismsoverAisthesubgroupAutf(M=A)ofAut(M=A)generatedbytheunionofallsubgroupsAut(M=M0)forallpossibleM0suchthatAµM0¹M.ItiseasytocheckthatAutf(M=A)isanormalsubgroupofAut(M=A).FromProposition3.4itfollowsthatitisalsoasubgroupof¡(M=A).Proposition3.7AssumethatN;N0arejTj+-resplendent,AµN¹N0,jAj·jTj,f2Aut(N=A),andfµ¹f2Aut(N0=A).Thenf2Autf(N=A)ifandonlyif¹f2Autf(N0=A).Proof.Assumethat¹f2Autf(N0=A)andchoose¹f1;:::;¹fn2Aut(N0=A)suchthat¹f=¹f1±¢¢¢±¹fnand¹fi2Aut(N0=Mi)forsomeMi¶A.ChooseamodelM¹NwithAµMandjMj·jTjclosedunderfandf¡1.EnlargethelanguageL(M)byaddingunaryfunctionsymbolsF1;:::;Fn;GandunarypredicatesU1;:::;Unandlet§bethesetofsentencesexpressing1.F1;:::;Fn;GareA-automorphisms;2.U1;:::;UnareelementarysubmodelscontainingA;3.Fi¹Ui=idUiforalli=1;:::;n;4.G=F1±¢¢¢±Fn;5.G(m)=f(m)forallm2M.§isconsistent.ByjTj+-resplendencythereisanexpansionofNMwhichsatises§.Thisgivesussomeg2Autf(N=A)suchthatf¹M=g¹M.Hencef±g¡12Aut(N=M)µAutf(N=A),thusf2Autf(N=A).ByasimilarargumentusingjTj+-resplendencyofN0,onecanshowthat¹fisstrongiffisstrong.Theorem3.8ForanyjTj+-resplendentmodelNandanyAµNsuchthatjAj·jTj,Autf(N=A)=¡(N=A):Proof.ByProposition3.4weknowthatAutf(N=A)µh[idN]¼Aiµ¡(N=A).Toshowthattheinclusion¡(N=A)µAutf(N=A)holdsitisenoughtocheckthatAutf(N=A)isaverynormalsubsetofAut(N=A).Letf2Autf(N=A)andletg2Aut(N=A)besuchthatf¼Ag.Forsome¹f;¹g2Aut(C=A)extendingf;grespectivelywehavethat¹f»A¹g.ByProposition3.7,¹f2Autf(C=A).SinceAutf(C=A)isanormalsubgroupofAut(C=A),¹g2Autf(C=A).AgainbyProposition3.7,g2Autf(N=A).c°2005WILEY-VCHVerlagGmbH&Co.KGaA,Weinheim 630E.CasanovasandR.Pel´aez:jTj+-resplendentmodelsandtheLascargroupCorollary3.9ForanyjTj+-resplendentmodelNandanyAµNsuchthatjAj·jTj,h[idN]¼Ai=¡(N=A).Proof.ByProposition3.4andTheorem3.8.Remark3.101.Itiseasytocheck(see[4,Proposition34])thatanautomorphismf2Aut(N=A)isin[idN]¼AifandonlyifN´Mf(N)forsomemodelM¶A.Inotherwords,forsomemodelM¶A,fhasanextensioninAut(C=M).2.Lascarin[4]pointedoutthatforstableT,[idN]¼Aitselfisagroup.Autf(N=A)=Aut(N=acleq(A))inastabletheory,anditsufcestotakeforthemodelMin1.arealizationofthenonforkingextensionoftp(N=acleq(A))overN.HenceAutf(N=A)=[idN]¼AforanyjTj+-resplendentmodelNofastablethe-oryT.3.IfTisano-minimaltheory,thenforeverymodelN¶A,Aut(N=A)=[idN]¼A.Thisisaconsequenceof[8,proofofLemma24]whereitisshownthatforeveryf2Aut(N=A)thereissomeextension¹f2Aut(C=A)offwhichistheidentityonsomemodelM¶A.Theorem3.11ForeveryjTj+-resplendentmodelNandeveryAµNsuchthatjAj·jTj,thequotientAut(N=A)=Autf(N=A)isindependentofthechoiceofN.Proof.WeprovethatAut(N=A)=Autf(N=A)»Aut(C=A)=Autf(C=A):Forthis,wedeneamapping±:Aut(N=A)¡!Aut(C=A)=Autf(C=A)choosingforanyf2Aut(N=A)anarbitraryextensionfµ¹f2Aut(C=A)andputting±(f)=¹fAutf(C=A).Itisclearlywelldenedanditisagrouphomomorphism.FromProposition3.7itfollowsthatitskernelisAutf(N=A).Wenishtheproofbyshowingthat±isonto.Letg2Aut(C=A).Weseeksomef2Aut(N=A)suchthat±(f)=gAutf(C=A).ChooseasubmodelM¹NsuchthatAµMandjMj·jTj.ByjTj+-saturationofNwemayndinNareal-izationM0oftp(g(M)=M).Theng(M)´MM0andthereissomeh2Aut(C=M)suchthath(g(M))=M0.M0isanelementarysubmodelofNcontainingAand(h±g)¹MisanA-isomorphismbetweenMandM0.WeenlargethelanguageL(M[M0)byaddingaunaryfunctionsymbolF.Let§bethesetofsentencesexpressing1.FisanA-automorphism;2.F(a)=h(g(a))foralla2M.§isconsistentandbyjTj+-resplendencythereissomef2Aut(N=A)suchthatf¹M=h±g¹M.Let¹f2Aut(C=A)beanarbitraryextensionoff.Thenh±g±¹f¡12Aut(C=M0)µAutf(C=A)andh2Autf(C=A).Thereforeg±¹f¡12Autf(C=A),thatis,±(f)=¹fAutf(C=A)=gAutf(C=A).4jTj+-saturatedstronglyjTj+-homogeneousmodelsTheorems3.8and3.11holdforaclassofmodelsstrictlywiderthantheclassofalljTj+-resplendentmodels.TheargumentsgivensofarcanberenedtoshowthattheyarealsotrueforjTj+-saturatedstronglyjTj+-homogeneousmodels.AspointedoutinProposition2.2,3.,alljTj+-resplendentmodelsarejTj+-saturatedandstronglyjTj+-homogeneous.Howeveritiseasytondnon-saturatedmodelsofstabletheories(evenofMorleyrank2)whicharejTj+-saturatedandstronglyjTj+-homogeneous.AfterProposition2.2,4.,itisclearthatthesemodelsarenotjTj+-resplendent.Wewillnowindicateshortlyhowtheproofsgivenintheprevioussectionscanbemodiedtoobtainthismoregeneralresult.OnekeypointisthatTheorems3.8and3.11dependbasicallyonwhatwecalltheextensionproperty.Denition4.1LetAµM.WesaythatMhastheextensionpropertyoverAifeveryf2Aut(M=A)whichhasanextension¹f¶finAutf(C=A)isalreadyinAutf(M=A).c°2005WILEY-VCHVerlagGmbH&Co.KGaA,Weinheim Math.Log.Quart.51,No.6(2005)/www.mlq-journal.org631Notethatitisalwaystruethatevery¹f2Aut(C=A)extendingsomef2Autf(M=A)isstrong.Proposi-tion3.7showsthatalljTj+-resplendentmodelshavetheextensionpropertyoversmallsubsets.WeshowthatthisisalsothecaseforjTj+-saturatedstronglyjTj+-homogeneousmodels.OurproofusesthesameideaastheonepresentedbyZieglerin[8,proofofCorollary3].Proposition4.2AssumeNisjTj+-saturatedandstronglyjTj+-homogeneous.ThenforeachAµNsuchthatjAj·jTj,NhastheextensionpropertyoverA.Proof.Letfµ¹f2Autf(C=A)andchoose¹f1;:::;¹fnsuchthat¹f=¹f1±¢¢¢±¹fnand¹fi2Aut(C=Ni)forsomeNi¶A.WemayassumethatjNij·jTjforeachi.ChooseamodelM¹NwithAµMandjMj·jTj.LetM0=MandMi+1=¹fi+1(Mi)fori=0;:::;n¡1.ObservethatM=M0´N1M1´N2M2´N3¢¢¢´Nn¡1Mn¡1´NnMn=¹f(M)=f(M):ByusingjTj+-saturationofN,choosenowmodelsN0i¹NandM0i¹NsuchthatM0;:::;Mn;N1;:::;Nn´Mf(M)M00;:::;M0n;N01;:::;N0n:ThenAµN0iandM=M00´N01M01´N02M02´N3¢¢¢´N0n¡1M0n¡1´N0nM0n=f(M):ByjTj+-stronghomogeneityofNwemaynownd,foreachi=0;:::;n¡1,gi+12Aut(N=N0i)suchthatgi+1(M0i)=M0i+1.Letg=g1±¢¢¢±gn.Theng2Autf(N=A)andg¹M=f¹M.Henceg±f¡12Aut(N=M)µAutf(N=A)andhencef2Autf(N=A).Theorem4.3ForanyjTj+-saturatedandstronglyjTj+-homogeneousmodelNandanyAµNsuchthatjAj·jTj,Autf(N=A)=¡(N=A)=h[idN]¼AiandAut(N=A)=Autf(N=A)isindependentofthechoiceofN.Proof.Fortherstassertion,observethattheproofgiveninTheorem3.7showsinfactthattheextensionpropertyoverAµNisenoughtogetAutf(N=A)=¡(N=A)=h[idN]¼Ai,andthenuseProposition4.2.TheproofoftheexistenceofanisomorphismbetweenAut(N=A)=Autf(N=A)andAut(C=A)=Autf(C=A)isaslightmodicationoftheproofgivenforTheorem3.11.Tocheckthat±isonto,insteadofenlargingthelanguagebyaddingF,usejTj+-stronghomogeneityofNtoobtainsomef2Aut(N=A)suchthatf¹M=h±g¹M.Nowlet¹f2Aut(C=A)beanarbitraryextensionoff.Thenh±g±¹f¡12Aut(C=M0)µAutf(C=A)andh2Autf(C=A).Thereforeg±¹f¡12Autf(C=A),thatis,±(f)=¹fAutf(C=A)=gAutf(C=A).AcknowledgementsThepresentworkwaspartiallysupportedbyresearchprojectsBFM2002-01034fromSpanishDGICYTand2002SGR00126fromCatalanDURSI.References[1]J.BarwiseandJ.S.Schlipf,Anintroductiontorecursivelysaturatedandresplendentmodels.J.SymbolicLogic41,531–536(1976).[2]E.Casanovas,Compactlyexpandablemodelsandstability.J.SymbolicLogic60,673–683(1995).[3]E.Casanovas,D.Lascar,A.Pillay,andM.Ziegler,Galoisgroupsofrstordertheories.J.Math.Logic1,305–319(2001).[4]D.Lascar,Onthecategoryofmodelsofacompletetheory.J.SymbolicLogic47,249–266(1982).[5]D.LascarandA.Pillay,Hyperimaginariesandautomorphismgroups.J.SymbolicLogic66,127–143(2001).[6]B.Poizat,CoursdeTh´eoriedesModeles(Nural-Mantiqwal-Ma'rifah,Villeurbanne1985).[7]S.Shelah,ClassicationTheory(NorthHolland,Amsterdam1978).[8]M.Ziegler,IntroductiontotheLascargroup.In:TitsBuildingsandtheModelTheoryofGroups(K.Tent,ed.).LondonMathematicalSocietyLectureNotesSeries291,pp.279–298(CambridgeUniversityPress,Cambridge2002).c°2005WILEY-VCHVerlagGmbH&Co.KGaA,Weinheim