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某些可解群的外自同构建的阶.docx

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某些可解群的外自同构建的阶 Introduction: Grouptheoryisabranchofmathematicsthatdealswiththestudyofgroups,whicharemathematicalobjectsthatformthebasisofmodernalgebra.Agroupisasetofelementsthatsatisfycertainproperties,suchasclosure,associativity,identity,andinverse.Groupsareusedinmanyareasofmathematics,physics,chemistry,computerscience,andengineering,tonameafew.Oneimportantconceptingrouptheoryisthenotionofanautomorphism,whichisabijectivemapfromagrouptoitselfthatpreservesthegroupstructure.Anautomorphismofagroupisanisomorphismfromthegrouptoitself.Inthispaper,wewillstudytheorderoftheexternalautomorphismgroupofagivengroup,alsoknownastheouterautomorphismgroup.Morespecifically,wewillfocusonthecasewherethegivengroupisasolublegroup,andshowthattheorderofitsouterautomorphismgroupisboundedbyafunctionofitsorder. Solublegroups: Asolublegroupisagroupthatcanbewrittenasasequenceofsubgroups,eachofwhichisnormalinthepreviousone,withfactorgroupsthatareallabeliangroups.Inotherwords,asolublegroupisagroupthatcanbedissolvedintosmaller,abelianpieces.Forexample,thegroupofintegersisasolublegroup,asitcanbewrittenasasequenceofsubgroups: {0}<{±1}<{±2}<{±3}<... whereeachsubgroupisnormalinthepreviousone,andthefactorgroupsareallisomorphictothecyclicgroupZ.Therearemanyinterestingexamplesofsolublegroups,suchasthedihedralgroups,thesymmetricgroups,thealternatinggroups,andtheSylowp-subgroupsoffinitegroups,tonameafew. Outerautomorphisms: AnouterautomorphismofagroupGisanautomorphismofGthatisnotaninnerautomorphism,i.e.,itcannotbewrittenasconjugationbyanelementofG.ThegroupofouterautomorphismsofGisdenotedAut(G)/Inn(G),whereAut(G)isthegroupofallautomorphismsofG,andInn(G)isthenormalsubgroupofAut(G)consistingofinnerautomorphismsofG.IfGisabelian,thenInn(G)isthetrivialgroup,andhenceAut(G)/Inn(G)isisomorphictoAut(G).However,ifGisnon-abelian,thenAut(G)/Inn(G)isanon-trivialgroupthatmeasurestheamountofsymmetryofGthatisnotcapturedbytheconjugationactionofGonitself. Theorems: LetGbeafinitesolublegroup.Thentheorderofitsouterautomorphismg