第1页共7页编号：时间：2021年x月x日书山有路勤为径，学海无涯苦作舟页码：第1页共7页ApplicationofTheFundamentalHomomorphismTheoremofGroupLIQian-qianLIUZhi-gangYANGLi-ying（DepartmentofMathematicsandComputerScience,GuangxiTeachersEducationUniversity,NanningGuangxi530001,P.R.China）Abstract:Thefundamentalhomomorphismtheoremisveryimportantconsequenceingrouptheory,byusingitwecanresolvemanyproblems.InthispaperweresearchesmainlyaboutthefundamentalhomomorphismtheoremappliedtodirectproductsofgroupsandgroupofinnerautomorphismsofagroupG.Keyword:TheFundamentalHomomorphismTheorem;DirectProducts;InnerAutomorphismsMR(2003)SubjectClassification:16WChineseLibraryClassification:O153.3Documentcode:AIntherealmofabstractalgebra,groupisoneofthebasicandimportantconcept,haveextensiveapplicationinthemathitselfandmanysideofmodernsciencetechnique.ForexampleTheoriesphysics,Quantummechanics,Quantumchemistry,Crystallographyapplicationareclearcertifications.Sothat,afterwestudyabstractalgebracourse,godeepintoagroundoftheoriesofresearchtohavethenecessityverymuchmore.Inthecontentsofgroup,thefundamentalhomomorphismtheoremisveryimportanttheorem,wecanuseitprovemanyproblemsaboutgrouptheory,inthispapertoproveseveralconclusionsasfollowwiththefundamentalhomomorphismtheorem:Thesecontentsareallstandardifwenottothespecialprovisionandexplained.Definition1.LetNbeasubgroupofagroupGwithsymbol≤G,wesayNisthenormalsubgroupoffifoneofthefollowingconditionshold.Tosimplifymatters,wewriteNf.(1)Na=aNforany;(2)aNa−1=Nwheneverany;(3)ana−1∈Nforeveryn∈Nandany.Definition2.ThekernelofagrouphomomorphismϕfromGtoagroup¯Gwithidentityeistheset第2页共7页第1页共7页编号：时间：2021年x月x日书山有路勤为径，学海无涯苦作舟页码：第2页共7页{x∈G|ϕ(x)=e}.Thekernelofϕisdenotedby.Definition3.Letbeacollectionofgroups.Theexternaldirectproductof,广西自然科学基金(0447038)资助项目writtenas,isthesetofallm-tuplesforwhichtheitscomponentisanelementof,andtheoperationiscomponentwise.Insymbols=,whereisdefinedtobeNoticethatitiseasilytoverifythattheexternaldirectproductofgroupsisitselfagroup.[4]Definition4.LetGbeagroupandHbeasubgroupofG.Foranya∈G,thesetaH={ah|h∈H}iscalledtheleftcosetofHinGcontaininga.AnalogouslyHa={ha|h∈H}iscalledtherightcosetofHinGcontaininga.Lemma1.[1](Thefundamentalhomomorphismtheorem)LetϕbeagrouphomomorphismfromGto¯G.ThentheN=isthenormalsubgroupoff,and.Tosimplifymatters,wecallthetheoremastheFHT.Lemma2.[2]LetϕbeagrouphomomorphismfromGto¯G.Thenwehavethefollowingproperties:（1）IfHisasubgroupofG,thenisasubgroupof¯G;（2）IfHisanormalinG,thenisanormalin¯G;（3）If¯Nisasubgroupof¯G,thenisasubgroupofG;（4）If¯Nisanormalsubgroupof¯G,thenisanormalsubgroupofGLemma3.[3]LetϕbeahomomorphismfromagroupGtoagroup¯G,and¯N¯G,.ThenG/N≃¯G/¯N.Lemma4.[4]LetHbeasubgroupofGandletbelongtoG,then:第3页共7页第2页共7页编号：时间：2021年x月x日书山有路勤为径，学海无涯苦作舟页码：第3页共7页（1）aH=Hifandonlyifa∈H;（2）aH=HaifandonlyifH=a−1Ha.Byusingtheabovelemmaswecanobtainthefollowingmainlyresults.Theorem1.LetGandHbetwogroups.SupposeJGandKH,thenand(G×H)/(J×K)≃(G/J)×(H/K).Proof.Firstwewillprove.Forany(g,h)∈(G×H)andevery(j,k)∈(J×K).Wehave:(g,h)(j,k)(g,h)−1=(g,h)(j.k)(g−1,h−1)=(gjg−1,hkh−1).SinceJ⊲GandK⊲H,wecanget(gjg−1,hkh−1)∈(J×K),i.e.(g,h)(j,k)(g,h)−1∈(J×K).Thus.WemakeuseoftheFHTtoprovethat(G×H)/(J×K)isisomorphicto(G/J)×(...
