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1、[toappearintheProceedingsoftheAmericanMathematicalSociety]EveryGroupHasATerminatingTransfiniteAutomorphismTowerJoelDavidHamkins†CityUniversityofNewYorkAbstract.Iterativelytakingtheautomorphismgroupofanygroupleads,trans-finitely,toafixedpoint.Theautomorphismtowerofagroupisobtainedbycomp
2、utingitsautomorphismgroup,theautomorphismgroupofthatgroup,andsoon,iteratingtransfinitely.Eachgroupmapscanonicallyintothenextusinginnerautomorphisms,andsoatlimitstagesonecantakeadirectlimitandcontinuetheiteration.G→Aut(G)→Aut(Aut(G))→···→Gω→Gω+1→···→Gα→···Thetowerissaidtoterminateifafi
3、xedpointisreached,thatis,ifagroupisreachedwhichisisomorphictoitsautomorphismgroupbythenaturalmap.Thisoccursifacompletegroupisreached,onewhichiscenterlessandhasonlyinnerautomorphisms.Inthespecialcasethattheinitialgroupiscenterless,matterssimplifycon-siderably:inthiscaseallthegroupsap
4、pearinginthetowerarecenterless(seeHulse[1970]),and,consequently,allthenaturalmapsareinjective.Thetowercanthereforebeviewedasbuildingupwardstolargerandlargergroups;thequestioniswhetherthisbuildingprocesseverstops‡.Wielandt[1939]provedtheclassicalre-sultthattheautomorphismtowerofanyce
5、nterlessfinitegroupterminatesinfinitelymanysteps.RaeandRoseblade[1970]provedthattheautomorphismtowerofanyarXiv:math/9808014v1[math.GR]4Aug1998centerlessCernikovgroupterminatesinfinitelymanysteps.Hulse[197ˇ0]provedthatthetheautomorphismtowerofanycenterlesspolycyclicgroupterminatesincoun
6、tablymanysteps(butnotnecessarilyafterjustωmanysteps).Solvingthe†MyresearchhasbeensupportedinpartbyagrantfromthePSC-CUNYRe-searchFoundation.IwouldliketothankbothDanielSeaboldandDanielVellemanforpointingoutasimplificationinmyproof.‡InHulse[1970],RaeandRoseblade[1970],andThomas[1985],th
7、etowerisonlydefinedinthisspecialcase;butthedefinitionIgivehereworksperfectlywellwhetherornotthegroupGiscenterless.Ofcourse,whenthereisacenter,onehashomomorphismsratherthanembeddings.2problemforcenterlessgroups,SimonThomas[1985]provedthattheautomorphismtowerofanycenterlessgroupeventual
8、lyterminates.Infact
