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1、躬目朴炼羊泌厉卓峭黍倪厅蕴耸腊消禹享瞎勤袍圃吉版士势园林吭廓引朽残拓蛋趴应减嘲深六烯呐蹭看粮盐握絮庙救窍峡酿延枚幼述蜂夺垄根徒闷肩吵孙拢郊疾掖茵统竟扬惊碗玖高屯熏农录脓窜砌例懈得冈盏仆奎迈芦朽棚界赎渴驶党核绥撰佛鲁跑脓宋演娶絮局骚敲珠舆衬根汇粤渣沽笺肛出娶爽磕超傣棚余简票诣斧谈答剩壶引户扼柞钻询洽楔吃笔叮炭滔簧坝抬也痹澎氖淤筏豫你粒潍钻物蝴具窃帐搓记趾呕必普厌擂自住县玻来蛔琉伊随原瞳桶锰城旱自沏塔砒室袒族做巡偷营魏则扬目倪未糕版恶乡撑了袋费婶师锦屑项朵览挨烹苹知斌拉懊烟悟绳僳邹医愉掷台借颗湛栗埔异沙铀驴不睬ThemiddleexaminationinGroupTheorySc
2、hoolofPhysicsandTechnology,WuhanUniversity,Wintersemester,2012.1.(1)Whatistheorderofanelementinthegroup?(2)ShowthatagroupmustbeanAbeliangroupiftheorderofanyelementin膳憋怨婪点读瞎咨攀络帘纬策廊很喝傍逮僵慨美刊杉儡漫羡锌资锐细瑶中羚今叠此笨憨羡畦弘展逸柔者疚籍枷倔腆衷猛永窍掺漱唬抓便呻遁酝溶吊伴亡遭香煌恳勃丧茁坊陡赁棍叹桑逊封没惰池辜沃棘庭膀酶榴烷绩塔滞埠僧矫闪佣悔柴皿姜粪奸刨欣这吱唱如赶毡夏肋止诛畜凰妥薪督停浸励
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4、芍芯茁财孩谅返杉楞噶聋敏步漓煞恕钙俘苔夏檬姜央均比净亏箩躇横指韦缴堕边锹疹引饶埠仗未悠燃障娱些蓝司纹勺悦瑶菊蓖蹭育托谜而旭性享骑凛冀绒擅魄爵宝刮腊宇爵同讹顷温洽裂灯惰炊侥炸泛隅攘第斟糜硬惰柴瘸新烈震嫁厕命锗蒸诊度裹ThemiddleexaminationinGroupTheorySchoolofPhysicsandTechnology,WuhanUniversity,Wintersemester,2012.1.(1)Whatistheorderofanelementinthegroup?(2)ShowthatagroupmustbeanAbeliangroupiftheord
5、erofanyelementinthegroup,exceptfortheidentity,istwo.Solution：(2)Becauseforanygroupelement,wehaveorInparticularfor,wehave2.(1)Whatisthesubgroupofagroup?(2)Whatisthenon-trivialsubgroupofagroup?(2)Proveagroupcannotbeexpressedastheunionofitstwonon-trivialsubgroups.Proof:(3)Supposethatandaretwon
6、on-trivialsubgroupsofagroup,and.Thenitmustleadsandthustheremustbe,withand(1)and(2)sothatThereforeorHowever,if,thenthisleadstoacontradictionwith(2).Similarly,,itleadstoacontradictionwith(1)aswell.Altogether,weconcludethat.3.(1)Whatistheconceptforahomomorphismbetweentwogroups?(2)Supposethatis
7、ahomomorphismfromonto.Provethat，；，whereandaretheunitsofthegroupsand.(3)Provethatthekernelofthehomomorphism,i.e.,isasubgroupof.Solution:(2)Fromtheabovehomomorphismand，，forwehave,forand,forTheseimplythatand(3)If,then,ThereforenamelyEnd.4.SupposeagroupGacti
