(第十一次群论作业)

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1、The11thAssignmentAssignmentaboutrotationgroupanddecompositionoftheself-directproductofgroupsProblem1Themultiplicationtableofthegroupisgivenasfollows.rightleftEABCFKMNEABCEABCAEMKBNEMCKNEFKMNNCBFKFCAMAFBFKMNFMKNKCFAMFABNBCFEBACBENMCNKEAMEK1，Writethecharactert

2、ableforthegroup.2，Ifweknowthattherepresentationmatricesofthegeneratorsandinatwo-dimensionalirreduciblerepresentationareandtwosetsofbasisfunctionsandrespectivelytransformaccordingtothistwo-dimensionalrepresentationofcombinetheproductfunctionsuchthatthebasisfu

3、nctionbelongstotheirreduciblerepresentationof.1.5特征表111111111-11-1112-20002.Problem2DefineamatrixbyWhereand,,arestandardPauli’smatrices.Foranarbitraryunitarymatrixasfollows5,(1)wecandefineatransformation,(2)forvector,obeyingfollowingrule(3)Then1,writetheexpr

4、essionofthetransformationmatrix,correspondingto,intermsofand.2,provethatisarealorthogonalmatrixand.3,searchforthreepairsofvaluesofandwhichcansatisfiesfollowingmappings4,calculatematrix,andproveisjusttherotationofarigidbodywithEuler’sangles,i.e.,.1.52故是实正交变换故

5、命题得证34.5==4.故与欧拉相同Problem3Provethatthereexistsahomomorphismmappingfromto,searchforthekernelofthishomomorphism.Whatisthefactorgroupofthishomomorphismmapping?对于任意5