It is easy to see that an inner automorphism of a group can always be extended to an inner automorphism of any group containing it Are inner automorphisms the only such automorphisms This is the problem of extensible automorphisms Here I discuss the ID: 22379
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1.2.Theoriginalproblemstatement.Thegeneralproblemstatementis:Problem1(Whichautomorphismsareextensible?).Anextensibleautomorphism(myownterminology)ofagroupisdenedasanautomorphismthatcanbeliftedtoanautomorphismforanyembeddingintoanothergroup.Whichautomorphismsofagroupareextensible?Thefollowingbasicfactsaredirect:Everyinnerautomorphism(thatis,conjugationbysomegroupelement)isextensible.Thus,thepropertyofbeinganinnerautomorphismisstrongerthanthepropertyofbeinganextensibleautomorphism.Thepropertyofextensibilityisclosedundercompositionandinversion.Thus,thesetofextensibleautomorphismsformsasubgroupofthegroupofallauto-morphisms.Itseems,fromexaminationofelementarycases,thatextensibleautomorphismsarepreciselytheinnerautomorphisms.Thisismyconjecture.1.3.Multipleextensions.Forconvenience,weusethenotation: where andarebothautomorphismproperties,toindicateasubgrouppropertywhereeveryautomorphismsatisfyingpropertyinthesubgroupcanbeliftedtoanautomorphismsatisfying inthewholegroup.Then,itistruethat:Innerautomorphism Innerautomorphismisheldforeverysubgroup,thatis,theabovepropertyisthetautology.Ontheotherhand,theproperty:Innerautomorphism Automorphismisequivalenttothesubgroupbeingfullynormalized(infrequentlyusedterminology):theWeylgroupofthesubgroup2isthewholeautomorphismgroup.Theproperty:Automorphism Automorphismisnottrueforallsubgroups.Thispropertyofsubgroupsiscalledtheautomorphismextensionproperty.(myownterminology)Thetrivialsubgroupandthegroupitselfsatisfytheautomorphismextensionproperty,buttherearemanysubgroupswhichdonot.Forinstance,thereareautomorphismsofZ=pZZ=pZthatdonotlifttoautomorphismsofthegroupZ=pZZ=p2Z(inwhichitisembedded).Notethatallfullynormalizedsubgroupshavetheautomorphismextensionproperty.Ontheotherhand,ifthepropertyontherightisstrengthenedsomewhat,weget:Automorphism ExtensibleautomorphismThisistrueforeverysubgroup.Thisisbecause,bydenition,anextensibleautomor-phismisonethatliftstoanautomorphismforeveryembedding. 2TheWeylgroupofasubgroupisthequotientofitsnormalizer(inthegroup)byitscentralizer(inthegroup),embeddednaturallyintheautomorphismgroupofthesubgroup.Ingeometriccontexts,itoftenarisesforselfcentralizingsubgroups(suchastori)inwhichcaseitbecomesthequotientofthenormalizerofthesubgroup2 2.4.GroupswithAbeliansplinchers.WhatarethegroupsthatpossessAbeliansplinchers?Reversingthequestion,constructallcentralfactorsofautomorphismgroupsofAbeliangroups,wheretheAbeliangroupischaracteristicinthesemidirectproduct.Clearly,inordertostudythisproblemitisnecessarytorstunderstandthestructureofautomorphismgroupsofAbeliangroups.Morespecically,wecanbeginbyunder-standingthestructureoftheautomorphismgroupsofAbeliangroupsofprimepowerorder.LetusbeginwiththeelementaryAbeliangroups.TheautomorphismgroupofanelementaryAbeliangroupoforderpkisthegenerallineargroupoforderkoverFp,typicallydenotedasGLn(Fp).Weprovethat:Claim5.EveryelementaryAbeliangroupischaracteristicinitssemidirectproductwithitsautomorphismgroup.Proof.ClearlytheelementaryAbeliangroupisaminimalnormalsubgroupofthesemidi-rectproduct(becausethereisalineartransformationmappinganysubspaceofittoan-othersubspacewiththesamedimension).Thus,ifthereisanyautomorphismthatdoesnottakeittoitself,itmusttakeittosomeotherminimalnormalsubgroupintersectingittrivially.Butthenthetwonormalsubgroupswouldcommuteelementwise,andweknowthatnoelementoutsidetheelementaryAbeliangroupcommuteswitheveryelementofit.So,everyautomorphismmusttaketheelementaryAbeliangrouptoitself.ThusGLn(Fp)possessesanAbeliansplincher.2.5.DirectproductsofAbelianandlineargroups.Combiningtheideasobtainedsofar,wecanconcludethat:Claim6.Adirectproductofgenerallineargroupsoverdi erentprimes,andanAbeliangroupwhoseorderisrelativelyprimetoallthegenerallineargroupsandtheirunderlyingprimes,hasanAbeliansplincher.2.6.Completegroups.Therearesomegroupsforwhichsplinchersarenotneededtoestablishthattheysatisfytheconditionsofproblem5.Thesearethegroupswhereeveryautomorphismisinner.Clearly,forsuchgroups,theproblemconditionsaretriviallysatised.Centerlessgroupswhereeveryautomorphismisinneraretermedcompletegroups.Inparticular,thesymmetricgroupsSnwheren6=2;6arecomplete.3.Furtherquestions3.1.Onsplinchers.Wehaveseenthefollowingproperties,eachimplyingtheonebelowit:(1)ThepropertyofhavinganAbeliansplincher(2)Thepropertyofhavingasplincher,orofbeingsplinchable(3)Thepropertyofsatisfyingtheconditionsofproblem5(4)Thepropertyofsatisfyingtheconditionsofproblem3(5)Thepropertythateveryextensibleautomorphismofthegroupisinner(6)Thepropertythatevery!extensibleautomorphismofthegroupisinnerConjecturescanbeformulatedatvariouslevels.WemayconjecturethatallgroupshaveAbeliansplinchers,whichisthestrongestpossiblestatement.Wemayconjecturethatallgroupsaresplinchable,whichissomewhatweaker.Wemayconjecturethatallgroupssatisfytheconditionofproblem5,whichisstillweaker.8 3.4.AnalogousquestionforthevarietyofAbeliangroups.ThequestionforthevarietyofAbeliangroupsisasfollows:\whataretheautomorphismsofanAbeliangroupthatlifttoautomorphismsofallAbeliangroupscontainingit?"Notethattheproblemwehavesolvedforthecaseofgroupsistheautomorphismsthatlifttoautomorphismsofallgroupscontainingit,whiletheproblemwenowconsiderisofautomorphismsthatextendtoautomorphismsofallAbeliangroupscontainingit.Inthiscase,themultiplicationmapsareconditionallyendomorphing,thoughnotcon-ditionallyautomorphing.Thisquestionisinterestingandpossiblyeasiertosolve.SomeprogresshasbeenmadeonitthatIamnotdiscussinghere.3.5.Generalhomomorphisms.Wehavesofarbeeninterestedindeterminingthoseautomorphismsthatgaverisetoautomorphismsforembeddingsintoothergroups.Em-beddingsarejustinjectivehomomorphisms.Thus,wehavelookedatautomorphismsthatgiverisetoautomorphismsoverinjectivehomomorphisms.Amoregeneralquestionmightbetondautomorphismsthatgaverisetoautomorphismsforallhomomorphisms.Thatis:Problem6.FindallautomorphismsofagroupG,suchthatforanyhomomorphismfromGtoagroupH,thereisanautomorphism0onHsuchthat0:=:.Ifwerequiretobeinjective,thenwegetpreciselythedenitionofextensibleauto-morphism.Thustheabovepropertyisapriorisomewhatstrongerthanthatofbeingextensible.Weshallcallautomorphismsthatsatisfytheconditionsoftheaboveproblemhomomorphismtransferable.(myownterminology)Itiseasytoseethatanautomorphismishomomorphismtransferableifandonlyifitisextensibleanditsatisestheaboveconditionforsurjectivehomomorphisms.Automorphismsthatsatisfytheconditionforsurjectivehomomorphismshallbecalledquotientableautomorphisms.(myownterminology)Clearly,innerautomorphisms,orautomorphismsarisingviaconditionallyautomorph-ingformulas,arehomomorphismtransferable.Whatcanwesayaboutquotientableautomorphisms?Indeed,therearequotientableautomorphismsthatarenotinner.Thisfollowsfromtheobservationbelow.Claim7.Anyautomorphismthatcanbedescribedbysome\algebraicformula"intermsoftheuniversalalgebraoperationsusingoneormoreparameters,isquotientable.Infact,theautomorphismonthequotientisgivenbythesamealgebraicformulawiththeparametersbeingtheimagesoftheoriginalparametersviathequotientmap.Callanautomorphismalgebraicallyformulable(myownterminology)ifitcanbeexpressedviaanalgebraicformula.Notethatthesetofalgebraicallyformulableauto-morphismsisclosedundercompositionthoughitisnotclearwhetheritisclosedunderinversion.Thisgivesrisetothenalproblemthatneedstobeexplored:Problem7.Givenagroup,whatcanbesaidaboutthecollectionofquotientableauto-morphisms?Iseveryquotientableautomorphismalgebraicallyformulable?4.Inspirationandacknowledgement4.1.Inspirationfortheproblem.TheproblemcametomethroughaformalismIhavedevelopedcalledpropertytheory,wherebygrouppropertiesandsubgrouppropertiesaresubjectedtomanipulationandstudiedunderthesemanipulations.Forinstance,expressingsubgrouppropertiesusingthe notation(introducedinsection1.3)helpsustouncoversomefactsaboutthem.10 4.2.Thepartialsolutionsofar.ThesolutionuncoveredsofarhaslargelybeenduetothekindhelpprovidedbyDr.I.M.Isaacs,whofurnishedtherstexampleofthenon-Abeliangroupoforder21,andalsoprovidedtheexampleofGL3(F2).Healsosuggestedsomeapproachestogeneralizingtheproblemandhissuggestionsledtomyformulatingproblems4and5.HegavetheproofforthecaseofcyclicgroupsandencouragedmetosettlethecaseforAbeliangroupsbeforetryingtoproceedfurther.Theconversionofthespecicproofs(fortheAbeliancase)intogeneralideasofsplinch-ersandcentralfactorswasdonebymetodeterminehowtheapproachcanbegeneralized.4.3.Alternativeapproaches.PriortodiscussingtheproblemwithDr.Isaacs,IalsodiscusseditwithProfessorRamanan,whotriedanalternativeapproachusingcharactertheory.However,theapproachmetwithsomeroadblocks.Theroughideawastousethefactthattwoelementsnotinthesameconjugacyclassdi erforsomeclassfunctionandhencewecanobtainacharacterwheretheytakedi erentvalues.Startingwiththisobservation,wetrytoconstructagroup,byaslightvariationofthegenerallineargroup,wherethereisnoautomorphismtakingoneelementtotheother.Ihaveworkedonthisapproachaswellwithoutmuchsuccess.However,ithasledmetoreformulatetheapproachinamannerthatcanbereconciledwiththesplincherapproach.Thiswillbediscussedlaterseparately.11 Indexalgebraicformulaconditionallyautomorphing,9algebraicformula,10automorphism,2!extensible,3kextensible,3algebraicallyformulable,10extensible,2homomorphismtransferable,10inner,1,2quotientable,10automorphismextensionproperty,2centralfactor,5conjugation,1extension,1groupAbelian,7complete,8cyclic,7elementaryAbelian,8holomorph,3lift,1semidirectproduct,5,8splincher,5StructureTheoremforAbeliangroups,7subgroupcharacteristic,5fullynormalized,2,3havingautomorphismextensionproperty,2minimalnormal,8normal,9transform,1Weylgroup,212