刘觉平群论第四次作业.doc

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1、Fillyournameandyourregistrationnumberintheaboveblanks!Pleaseusethe2003Word,TimesNewRomanforEnglishletters,or宋体（五号）forChinese,andmathtypeforequations(usethesizeforlettersof10.5,asconvention)towriteyourhomework,andgivetheelectronicfilesofyourhomeworkintimetothecorrespondingpersonbe

2、ingresponsibleforevaluating.ThedeadlineforsendingyourhomeworkthroughemailisSunday24:00oftheweekwhenthehomeworkisgiven.NextlectureinGroupTheorywillbegivenon12thOct.2013.Exercise4inGroupTheoryExercises4-1.SupposethatHisasubgroupofGwithindex.ProvethatHisainvariantsubgroupofG.解：∵指数∴有

3、2个不同的左陪集，一个为，一个为，若，则若，则∴是的不变子群。Exercises4-2.Provethattheintersectionoftwoinvariantsubgroupsisaninvariantsubgroup.证明：令,则是的子群，有∵是不变子群，∴∴∴是的不变子群Exercises4-3.Provethat(a)Alltheelementsofthesameconjugacyclassareofthesameorder.(b)Ifthenbecauseisconjugateto.证明：(a)设属于同一个共轭类，则有令的阶为，则有∴有相同

4、的阶（b）∵与共轭∴有∵∴即Exercises4-4.ThemultiplicationtableofthefinitegroupGisasfollows.EABCDFIJKLMNEABCDFIJKLMNEABCDFIJKLMNAEFIJBCDMNKLBFAKLEMNIJCDCILAKNEMJFDBDJKLAMNEFIBCFBEMNAKLCDIJICNEMLAKDBJFJDMNEKLABCFIKMJFIDBCNELALNIJFCDBEMAKMKDBCJFILANENLCDBIJFAKEMa)FindtheinverseofeachelementinG;b

5、)PointoutwhichelementscancommutewithanyelementinG;c)ListtheperiodandorderofeachconjugacyclassofG;Theorderofanelementgisthesmallestvalueofksuchthatgk=e.theperiodisthecyclicsubgroupgeneratedbysuchanelementg.d)FindtheelementsineachconjugacyclassofG;e)FindallinvariantsubgroupsinG.For

6、eachinvariantsubgroup,listitscosetsandpointoutontowhichgroupitsquotientgroupisisomorphic;f)MakeajudgmentwhetherGisisomorphicontothetetrahedralsymmetricgroupT,orisomorphicontotheregularsix-sidedpolygonsymmetricgroup.解：a)b)c,d)∵∴由Exercises4-3可知共轭类有，，代表元为，，代表元为，，代表元为，，代表元为，，代表元为e)不变

7、子群有陪集为，，，，，，，，，，陪集为陪集为，,,,商群同构的是f)与不同构，与不同构