广西自然科学基金资助项目

ApplicationofTheFundamentalHomomorphismTheoremofGroupLIQian-qianLIUZhi-gangYANGLi-ying（DepartmentofMathematicsandComputerScience,GuangxiTeachersEducationUniversity,NanningGuangxi530001,P.R.China）Abstract:Thefundamentalhomomorphismtheoremisveryimportantconsequenceingrouptheory,byusingitwecanresolvemanyproblems.InthispaperweresearchesmainlyaboutthefundamentalhomomorphismtheoremappliedtodirectproductsofgroupsandgroupofinnerautomorphismsofagroupG.Keyword:TheFundamentalHomomorphismTheorem;DirectProducts;InnerAutomorphismsMR()SubjectClassification:16WChineseLibraryClassification:O153.3Documentcode:AIntherealmofabstractalgebra,groupisoneofthebasicandimportantconcept,haveextensiveapplicationinthemathitselfandmanysideofmodernsciencetechnique.ForexampleTheoriesphysics,Quantummechanics,Quantumchemistry,Crystallographyapplicationareclearcertifications.Sothat,afterwestudyabstractalgebracourse,godeepintoagroundoftheoriesofresearchtohavethenecessityverymuchmore.Inthecontentsofgroup,thefundamentalhomomorphismtheoremisveryimportanttheorem,wecanuseitprovemanyproblemsaboutgrouptheory,inthispapertoproveseveralconclusionsasfollowwiththefundamentalhomomorphismtheorem:Thesecontentsareallstandardifwenottothespecialprovisionandexplained.Definition1.Letbeasubgroupofagroupwithsymbol≤,wesayisthenormalsubgroupofifoneofthefollowingconditionshold.Tosimplifymatters,wewrite.(1)forany;(2)wheneverany;(3)foreveryandany.Definition2.Thekernelofagrouphomomorphismfromtoagroupwithidentityistheset.Thekernelofisdenotedby.Definition3.Letbeacollectionofgroups.Theexternaldirectproductof,广西自然科学基金(0447038)资助项目writtenas,isthesetofallm-tuplesforwhichtheitscomponentisanelementof,andtheoperationiscomponentwise.Insymbols=,whereisdefinedtobeNoticethatitiseasilytoverifythattheexternaldirectproductofgroupsisitselfagroup.[4]Definition4.Letbeagroupandbeasubgroupof.Forany,thesetiscalledtheleftcosetofincontaining.AnalogouslyiscalledtherightcosetofHincontaining.Lemma1.[1](Thefundamentalhomomorphismtheorem)Letbeagrouphomomorphismfromto.Thenthe=isthenormalsubgroupof,and.Tosimplifymatters,wecallthetheoremastheFHT.Lemma2.[2]Letbeagrouphomomorphismfromto.Thenwehavethefollowingproperties:（1）Ifisasubgroupof,thenisasubgroupof;（2）Ifisanormalin,thenisanormalin;（3）Ifisasubgroupof,thenisasubgroupof;（4）Ifisanormalsubgroupof,thenisanormalsubgroupofLemma3.[3]Letbeahomomorphismfromagrouptoagroup,and,.Then.Lemma4.[4]LetHbeasubgroupofGandletbelongtoG,then:（1）ifandonlyif;（2）ifandonlyif.Byusingtheabovelemmaswecanobtainthefollowingmainlyresults.Theorem1.LetGandHbetwogroups.SupposeJGandKH,thenand.Proof.Firstwewillprove.Foranyandevery.Wehave:.Sinceand,wecanget,i.e..Thus.WemakeuseoftheFHTtoprovethatisisomorphicto.Thereforewemustlookforagrouphomomorphismfromontoanddeterminethekernelofit.Infactonecandefinecorrespondencedefinedby.Clearly,,theremustbetosatisfy.Thus,isonto.BecauseofJG,wehavefor,similarly,for.When,thereare.Forany,wehave====.Hence.Thereforeisgroupahomomorphismfromontoandistheidentityof.Forany,then,accordingtothepropertyofcoset,wecanget:ifandonlyifand,i.e.=.Nowletwelookatourproof:,isagrouphomomorphismfromontoandthekernelofis.AccordingtotheFHT,wecanget.Theorem2.Letisagrouphomomorphismfromonto.Ifand,thenwhere.Proof:AccordingtoLemma2.[2](2),weknow.Toestablish,wefirstlyneedtoconstructamappingandproveisagrouphomomorphismfromonto.Wegivethemappingdefinedbywhere=.For,sinceisasurjectionfromto,wemustbefoundsuchthat.Thusisonto.Forarbitrary,Thereforeisagrouphomomorphism.Wewillnowshow,infactweknowthatisidentityof,accordingtoLemma4,wecangetthatfor,then,say,sothat.Ontheotherhand,,,thatistosay,.Moreover,becauseof,therefore.Thatis.AccordingtotheFHT,wecanobtain. Theorem1andTheorem2applyExercise1andExercise2.Exercise1.isnormalsubgroupof,isanormalsubgroupof.Sothatforanyand,forafunction:wehaveisagroupisomorphism,sothatAssumeandaresetsofallthenonzerorealnumbersandpositiverealnumbersrespectively,itisreadilytoverifythattheyareindeedgroupwithordinarymultiplication.Exercise2.Letbegenerallineargroupof2×2matricesoverunderordinarymatrixmultiplication.Thenthemappingisagrouphomomorphismfromonto.Thegroupofmatriceswithdeterminant1overisanormalsubgroupof.Moreover.Definition5.Anautomorphismofgroupisjustagroupisomorphismfromtoitself.Thesetofallautomorphismsofgroupisdenotedby.Forany,iscalledaninnerautomorphismofandisthesetofallinnerautomorphismof.Theorem3:Letbeagroupandthemappingdefinedby.Then≤and.Proof.Itisclearlythat≤[5].Toshow,sufficeittoprovethatisanautomorphismofforany.1)(one-to-one)Forany,if=,thenbyusingcancellationlawofgroup.Thusisone-to-one.2)(onto)Forany,wetake,then,sothatisonto.3)(O.P.)Forany,wehave.Thereforeisisomorphismfromto.Accordingtothedefinitionofautomorphism.Weknowisanautomorphismof.Noticethatforany,wehaveand.Infactforany,itisclearly.Also,Thus.Since,say,wehaveknown.Wecanobtain,i.e..Hencetheproofof≤iscomplete.Itiseasytoseethatforevery,ifandonlyifwhereisthecenterof(shortfor).Letbethemappingdefinedby,wewillprovethatisagrouphomomorphismfromGontoI(G)andthatCisitskernel.Forevery,wecanreadilyfindthat,thatistosay,isonto.Forany,since,sothatisagrouphomomorphismfromonto.Noticethatforanyandevery,wehave,i.e.,,thatis.Weobtain,hence.Next,forany,wekno