刘觉平群论第二次作业.doc

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1、Exercise2inGroupTheoryPleasegivetheelectronicfiles(WORD2003document)ofyourhomeworkintimetothecorrespondingpersonbeingresponsibleforevaluating.DeadlineforsendingyourhomeworkthroughemailisSunday24:00oftheweekwhenthehomeworkisgiven.Problem1.ShowthatagroupmustbeanAbeliangroupiftheorderofanye

2、lementinthegroup,exceptfortheidentity,is2.证明：对于任意群都有元素令即又有得故群是阿贝尔群Problem2．(Lorentzgroup)ProvethatinMinkowskispace-timewherethefourcoordinatesofaneventisexpressedas,alltheLorentztansformationsalongwiththeX-axis(namelyallboostsalongwithX-axis)oftheformwhere,,formagroup.证明：取取（1）封闭性：取，则，显然且

3、，所以得证。（2）结合律：由（1）中结论，易知有（3）当时即有（4）当时综上所述，该变换构成一个群。Problem3.LetGbeafinitecyclicgroup,andletnbeapositiveintegerwhichdividestheorderofG,.ProvethatGhasacyclicsubgroupofGofordern,.证明：取为生成元，生成n阶循环群，必为的子群，故是的n阶循环子群Problem4.Let.Letx,ybethepermutationsinwhicharegivenby,,andletKbethesubgroupof.Sho

4、wthatthefunction,definedby(),isahomomorphism.(Notation：meansthesubgroupgeneratedbythegroupelementsxandy.)证明：要证明该映射是一个同态，只要证明由得所以左边=右边故成立所以()是一个同态。