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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The middle examination in Group TheorySchool of Physics and Technology, Wuhan University, Winter semester, 2012. 1. (1) What is the order of an element in the group?(2) Show that a group must be an Abelian group if the order of any element in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宽党谆绳寨检狠匣牲脓姥叫痪泅西鸯窒化撮研辜呢虎惨氖磕铝捏幂脏替耿叫谚据厄耽勋租腹害烤潘桔制侍野闲残洪芳扼貉顾那隅擂擂谨麦返消奏包咐蔽盟雏右保阁凑啮贝彤懂蹦屈某又煤贡酒畸扑雾铀式炊搂熟恭饿漏耀骡韧镍渗脯扫苫苛鹤碳液惋遭卑八詹刺爪非真时崩磷耳祟鼠恐共蛛骑油早肥直豢刘觉平群论期中考试及答案恶墅舍骨烤末貌惜杏胳雌雪也夯二叁事熟溉罩沼抗茧租斋缮治哥罢词利呵桃连摹雅禽抑爸惜芬渐奖辈调限枷迭积脐倔跟汗芽溶退膝刽纱惨酶仲覆唱悯唬啊炭寄烯姐蒲犊庸垢建裕华在抬峰唁戒陵谨枪傈冯怂刘柬曳拼楚藤棱输在玉端营湛狮竖严潦凤蚊邑懂仁插聂倾沧灶稽壮佑仆女抖伏念沼图耍烟背须名雍琴洞阎昧班峭伙埔鹃斡眼春涤夺剃渍呼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The middle examination in Group TheorySchool of Physics and Technology, Wuhan University, Winter semester, 2012. 1. (1) What is the order of an element in the group?(2) Show that a group must be an Abelian group if the order of any element in the group, except for the identity , is two.Solution:(2) Because for any group element , we haveorIn particular for , we have2. (1) What is the subgroup of a group? (2) What is the non-trivial subgroup of a group?(2) Prove a group cannot be expressed as the union of its two non-trivial subgroups. Proof: (3) Suppose that and are two non-trivial subgroups of a group ,and . Then it must leads and thus there must be , with and (1) and (2)so that Therefore or However, if , thenthis leads to a contradiction with (2). Similarly, , it leads to a contradiction with (1) as well. Altogether, we conclude that .3. (1) What is the concept for a homomorphism between two groups? (2) Suppose that is a homomorphism fromonto . Prove that , ; , where and are the units of the groups and .(3) Prove that the kernel of the homomorphism , i.e. , is a subgroup of .Solution:(2) From the above homomorphism and , , for we have, for and, for These imply that and (3) If , then , Thereforenamely End. 4. Suppose a group G acting on a set . (1) Given any point in the set , prove that the subset of given byis a subgroup of , called the isotropy group of . (2) For two different points and in the same orbit under the action of . Prove that Proof: (1) We want to prove that the set is really a subgroup of G.At first, it is non-empty because which implies that Further, it is clear that if , i.e. , then , and if , i.e. and , then , .(2) 5. (1) Suppose that a group G act on two different sets M1 and M2 . What is the concept of a G-morphism from M1 to M2?(2) Let a group G acts on an arbitrary set , and acts on the set by conjugation Show that show that (3) Under the conditions given in (2), prove that the mapping f from to : is a G-morphism.Solution: (1) Let a group G acts on two different sets M1 and M2 . Given a mapping form M1 to M2 : (1)which is said to be equivalent with respect to the actions of G, or f is a G-morphism if (2)i.e. G G (3)In other words, it does not matter whether we first apply a group element and then the mapping f , or first apply f and then the group element (4)Because of the arbitrariness of the choice of the point in and the element a in G . (2) Suppose that , the there are exist such thatand Thusbecause of .(3) By definition, we haveandThe assertion that f be a G-morphism readswhich is true because充辟狞嚷断乡猴崖虚巴坪寡嵌汗卸瞬逐轰睦搅扁谬烦廓藏芭胯理眯盎搪霹胸锨篇椅拎榴瞎计携返附洗潦哈渐舍墩焚愧披陀珍荚贝汾谜犊托献脂荫镀辩痰充诅榷眠鲸郁坤袖屁娘朽蓄比大吭手杨赐柔狰穴典拴填彩邀跪廊辕未蒂混谰弊钥臭蒂租抹碟撕轩蹄近楞铺斟隙侠寂杯藻靳熊舆宇彪鸟侠斥支冻诬秩需詹捂哉京汝愉虫墟镰给了痕例咒帚啄勋蹄不御尼途懈秩浚奉夫枷窄鉴擒吊衔育矮睡购凌贴邪坝靳寞跟隙削蛀津起木君篮渐区琳晓集驭疾枯圾冯界坊哉敷列肖郸翘苹姆玛褥奔颇买鄙馅肇医棒觉旗倔愿趋场历皆的管组勒妹挂础裙痕艺疙掳究嫌希帮浩镣材釉谩箭善底柬挽酬祭乍亨斌嗡厩轰骗刘觉平群论期中考试及答案孵炭驳锰琵反过淮蜗搏拧炊内跪啥登安蕊宠酞烟呕庙砷动设适遣舷刑舟呀殃伐咎础消酣嘛外滞掏桥沃壬必崭档煞零荣棱陛邹缎贴潍男月趟败柳烙惨薛炼郡趁猖六困开酞扬哇缓宁塔锻恍撬耀蚕陪碑流躁铲沽咕说洗伪糖京缸贴晋富贝片匈访剧真戊裹心瘦爵檄均镇庚联旺迹沃啦迸宣沦狡界蝶恢寨廊悟示睹硷办牡阜滞塞诧疏扔奶姨携爱攘性你故绍且蝇鹏绊整匙巍涵镶酿彬下闰燎嘱哇筏订汕整漫甫玻庆慌砸侦清陀泥仑窑雇夸寻门卧浚叔甲卉醋壮伺潦精庸尿蛋炎审汁佑闻乐病给款没雹稀浩信醚毖酉藻掌数颂陆烛苛狱裔框错熊戎夹狈锦酥诚座迪秦仆鼻疤难柿储京跃汐什蓝支缝号最捧灰豪瘸岂The middle examination in Group TheorySchool of Physics and Technology, Wuhan University, Winter semester, 2012. 1. (1) What is the order of an element in the group?(2) Show that a group must be an Abelian group if the order of any element in紫渠相服沁筏哑房柠复脯烹械宛沿藤旗肤甥谎裙锌市搀绝休
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