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1、ThemiddleexaminationinGroupTheorySchoolofPhysicsandTechnology,WuhanUniversity,Wintersemester,2012.1.(1)Whatistheorderofanelementinthegroup?(2)ShowthatagroupmustbeanAbeliangroupiftheorderofanyelementinthegroup,exceptfortheidentity,istwo.Solution：(2)Becauseforan
2、ygroupelement,wehaveorInparticularfor,wehave2.(1)Whatisthesubgroupofagroup?(2)Whatisthenon-trivialsubgroupofagroup?(2)Proveagroupcannotbeexpressedastheunionofitstwonon-trivialsubgroups.Proof:(3)Supposethatandaretwonon-trivialsubgroupsofagroup,and.Thenitmustlea
3、dsandthustheremustbe,withand(1)and(2)sothatThereforeorHowever,if,thenthisleadstoacontradictionwith(2).Similarly,,itleadstoacontradictionwith(1)aswell.Altogether,weconcludethat.3.(1)Whatistheconceptforahomomorphismbetweentwogroups?(2)Supposethatisahomomorphismf
4、romonto.Provethat，；，whereandaretheunitsofthegroupsand.(3)Provethatthekernelofthehomomorphism,i.e.,isasubgroupof.Solution:(2)Fromtheabovehomomorphismand，，forwehave,forand,forTheseimplythatand(3)If,then,ThereforenamelyEnd.4.SupposeagroupGactingonaset.(1)Givenany
5、pointintheset,provethatthesubsetofgivenbyisasubgroupof,calledtheisotropygroupof.(2)Fortwodifferentpointsandinthesameorbitundertheactionof.ProvethatProof:(1)WewanttoprovethatthesetisreallyasubgroupofG.Atfirst,itisnon-emptybecausewhichimpliesthatFurther,itisclea
6、rthatif,i.e.,then,andif,i.e.and,then,.(2)5.(1)SupposethatagroupGactontwodifferentsetsM1andM2.WhatistheconceptofaG-morphismfromM1toM2?(2)LetagroupGactsonanarbitraryset,andactsonthesetbyconjugationShowthatshowthat(3)Undertheconditionsgivenin(2),provethatthemappi
7、ngffromto:isaG-morphism.Solution:(1)LetagroupGactsontwodifferentsetsM1andM2.GivenamappingformM1toM2:(1)whichissaidtobeequivalentwithrespecttotheactionsofG,orfisaG-morphismif(2)i.e.GG(3)Inotherwords,itdoesnotmatterwhetherwefirstapplyagroupelementandthenthemappi
8、ngf,orfirstapplyfandthenthegroupelement(4)BecauseofthearbitrarinessofthechoiceofthepointinandtheelementainG.(1)Supposethat,thethereareexistsuchthatandThusbecauseof.(3)Bydefinition,
