how tall is the automorphism tower of a groupnew

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1、[TalkfortheMathematicalSocietyofJapaninOsakaonOctober2,1998]Howtallistheautomorphismtowerofagroup?JoelDavidHamkinsKobeUniversityandTheCityUniversityofNewYorkhttp://www.library.csi.cuny.edu/users/hamkinsTheautomorphismtowerofagroupisobtainedbycomputingitsautomorph

2、ismgroup,theautomorphismgroupofthatgroup,andsoon,iteratingtransfinitely.Eachgroupmapscanonicallyintothenextusinginnerautomorphisms,andsoatlimitstagesonecantakeadirectlimitandcontinuetheiteration.G→Aut(G)→Aut(Aut(G))→···→Gω→Gω+1→···→Gα→···Thetowerissaidtoterminatei

3、fafixedpointisreached,thatis,ifagroupisreachedwhichisisomorphictoitsautomorphismgroupbythenaturalmap.Thisoccursifacompletegroupisreached,onewhichiscenterlessandhasonlyinnerautomorphisms.Thenaturalmapπ:G→Aut(G)istheonethattakesanyelementg∈Gtotheinnerautomorphismig,

4、definedbysimpleconjugationig(h)=ghg−1.Thus,thekernelofπispreciselythecenterofG,thesetofelementswhichcommutewitheverythinginG,andtherangeofπispreciselythesetofinnerautomorphismsofG.Bycomposingthenaturalmapsateverystep,oneobtainsacommutingsystemofhomomorphismsπα,β:G

5、α→Gβforα<β,andthesearethemapswhichareusedtocomputethedirectlimitatlimitstages.arXiv:math/9808094v1[math.LO]21Aug1998Muchofthehistoricalanalysisoftheautomorphismtowerhasfocusedonthespecialcaseofcenterlessgroups,forwhentheinitialgroupiscenterless,matterssimplifycon

6、siderably∗.Aneasycomputationshowsthatθ◦ig◦θ=iθ(g)foranyautomorphismθ,andfromthisweconcludethatInn(G)⊳Aut(G)and,forcenter-lessG,thatCAut(G)(Inn(G))=1.Inparticular,ifGiscenterlessthensoalsois∗InHulse[1970],RaeandRoseblade[1970],andThomas[1985],thetowerisonlydefinedi

7、nthisspecialcase;butthedefinitionIgivehereworksperfectlywellwhetherornotthegroupGiscenterless.Ofcourse,whenthereisacenter,onehashomomorphismsratherthanembeddings.2Aut(G),andmoregenerally,bytranfiniteinductioneverygroupintheautomor-phismtowerofacenterlessgroupiscent

8、erless.Inthiscase,consequently,allthenaturalmapsπα,βareinjective,andsobyidentifyingeverygroupwithitsimageunderthecanonicalmap,wemayviewthetowerasbuildingupward