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第七章群论§1 群的基本概念和一般理论一、群的定义和例子群是按照某种规律互相联系着的一些元素的集合，我们用G来表示这个集合，并设它含有的元素是A，B，C，E等等。
不是随便什么样的元素集合都构成群，要组成数学群必须满足下列四个条件：1.封闭性G中任何两个元素相“乘”（包括一个元素本身“平方”），其结果任然是G中的元素。
如A属于G：B属于G：则有()（7.1-1）“乘”这个术语是通用的说法，在这里它含有比初等代数里的“乘”更广泛的意义，也许用“组合”来代替更恰当一些，我们将在下面通过几个例子来阐明。
一个数学群必须首先定义一种乘法。
2.缔合性三个以上的元素相乘满足乘法的结合律。
如A B C=A ( B C )= (A B ) C (7.1-2)即在保持三个元素相乘先后次序一定的前提下，其结果与哪两个元素相乘无关。
3.单位元素G中有一个元素E，它同每一个元素相乘，都等于该元素本身，即E A=A E=A,(7.1-3)称E为单位元素或恒等元素。
4.逆元素G中每一个元素A，都有另一个元素A-1，两者相乘等于单位元素E，即A=A=E,(7.1-4) 称为的逆元素。
逆元素可以是该元素本身。
下面我们举几个群的例子（2）G={所有大于0的实数}集合G包含所有大于0的实数，对普通的乘法而言，组成一个群。
满足封闭性和缔合性是显然的。
1是单位元素，任一实数m的逆元素为。
(3) G={0,±1, ±2, ±3……±n…}集合G包含0和所有正负整数，对于加法而言，组成一个群，成为整数加群。
此例中“乘”的意思是加。
1+2=3 封闭性满足1+2+3=1+(2+3)=(1+2)+3=6 缔合性满足0+3=3+0=3 0是单位元素n+(-n)=0 n有逆元素-n 213（4）G={E、I} ( C i )这个群（称为C i）里面的二个元素是“对称操作”，E是不动，I为对原点的倒反。
这种群（组成元素是一些对称操作）称为对称群或点群。
第一章第一章 抽象群概论§1 什么是群什么是群？？群公理不同元素的集合不同元素的集合，，赋予一定的合成规则赋予一定的合成规则（（称为群称为群““乘法乘法””—— 加、乘、对易子等对易子等）。
）。
满足下列满足下列条件条件（（群公理群公理））： （1）封闭性 i g 和G g j ∈，则G g g g k j i ∈=⋅； （2）结合律 )()(k j i k j i g g g g g g ⋅⋅=⋅⋅；（3）存在唯一的单位元素e （或E ）G ∈ ，对任一元素j g 有j j j e g g e g ⋅=⋅=； （4）对每一元素有逆元对每一元素有逆元，，对i g 有 1−i g ，使e g g ii =⋅−1。
阶 —— 群元的个数群元的个数：：阶有限为有限群阶有限为有限群；；阶无穷为无限群阶无穷为无限群。
无限群又分无限离散和无限连续无限群又分无限离散和无限连续。
注：1. 乘法不可对易乘法不可对易，，即i j j i g g g g ⋅≠⋅。
若可对易若可对易，，则称为阿贝尔称为阿贝尔（（Abel ）群。
2. 若G c b a ∈,,，则G 中包含p l k c b a ,,（其中p l k ,,为整数为整数））。
例1．复数1，i ，-1，-i 组成四阶群组成四阶群。
四阶循环群 —— 由一个元素由一个元素，，i （或-i ）出发出发，，由它及其幂由它及其幂次次生成整个群G ，称为循环群称为循环群。
循环群必是阿贝尔群环群必是阿贝尔群。
n 阶循环群可表为{23,,...n a a a a e =}。
例2．所有实数组合所有实数组合，，加法运算下成群加法运算下成群。
全体正实数在乘法运算下成连续群全体正实数在乘法运算下成连续群。
例3．定轴转动定轴转动：：Π<Θ≤20，)2(SO 无限连续群无限连续群。
特例 —— 转角为m 倍nπϑ2=构成n 阶群n C ；定点转动定点转动（（三维空间转动三维空间转动））：),,(γβαR ，)3(SO 群。
群论及其应用绪论(一) 指导思想:学以致用, 理论与应用结合(二) 教学方式:讲课, 提问, 讨论, 习题, 自学(三) 讲义和参考书一, 讲义: 1, 内容跟不上发展2, 印刷质量差二, 参考书:(1) 基础部分: 任何一本群论方面的书(2) 应用部分: 杂志中发表的有关论文(3) 参考书目: 1, “群论基础教程”，侯云智编著，山东大学出版社 2, “群论及其在物理学中的应用”, 谢希德编著, 北京科学出版社; 3, “物理学中的群论”, 陶瑞宝编著, 上海科学技术出版社;4, “群论对分子振动的应用”, 赵择卿译, 高等教育出版社;5, “物理学中的群论”, 马中騏编著, 科学出版社;6, “群论及其在固体物理中的应用”, 徐婉棠等编著, 高等教育出版社;7, “ Elements of Group Theory for Physicists”,A.W. Joshi, John Wiley and Sons, New York;8, “Group Theorp in Physics”, Singapore National Printers, Ltd; 9, “Group Theory Application to Molecular Vibrations”,P. G. Puranik, S. Chand and Company Ltd . *questions on Galois and his Group1, When was Galois born ?(A) in 1821, (B) in 1911, (C) in 1811, (D) in 18122, How old was Galois when he first suggested the mathematical concept “group” ?(A) 16, (B) 18, (C) 20, (D) 283, What kind of talents was the famouse Ecole polytechnique in Paris for ?(A) scientiests, (B) artists, (C ) statemen, (D) enterprisers 4, How many times were the papers containing Galois’s important discoveries submitted and lost ?(A) 2, (B) 3, (C) 5, (D) 15, What did Poason think of Galois’s thought ?(A) clear, (B) great, (C) confused, (D) advanced6, What kind of man was Galois challenged by in a duel ?(A) teacher, (B) lawyer, (C) docter, (D) policeman7, How long did Galois spend for writing down his ideas on group before duel ?(A) an entire day, (B) an entire night,(C) part of a day, (D) part of a night8, Who announced Galois’s paper written at the night before death ?(A) his father, (B) his wife, (C) his friend, (D) his brother9, Who discovered and recognized Galois’s paper on Group ?(A) Cosh, (B) Poason, (C) Newwig, (D) Eienstein 10, Which kind of field in mathematics does Group belong to ?(A) biology, (B) quantum mechanics, (C) geomitry, (D) algebraGalois and his goupEvariste Galois was a brilliant young French mathematician. He was born in 1811. He firstly suggested the mathematical concept “ group” when he only was 18 years old in 1830.Although he was a mathematical genius, his abilities were not recognized by his teachers. In fact he was twice denied admission to the famous Ecole polytechnique, the school for mathematicians and scientists in Paris.Twice he submitted papers containing his important discovers to the Frenth Academy. However famous mathematicians Poason and Cosh had not understood his papers. Moreover Poason said that the author’s thought was confused. The papers were twice lost.At the age of 20, he became involved in a quarrel over awoman and was challenged to a dual. He spent the entire night before the dual writing down his ideas. The next morning he was killed. Someone said that the death of Galois was the result of a political intrique. Golois objected French monarchy. Practically he was killed by a king’s policeman.A friend of Galois annouced his paper written down at the night before death on an unimportant magazine and the paper was discovered and recognized by the other mathematician Newwig. It is Galois’s group.Galois studied groups in order to solve certain problems in algebra. His discoveries greatly expanded field of algebra. Furthermore, his ideas have also been applied to physics and chemistry, for example, quantem mechanics, atomic physics, solid state physics etc, and it start to be applied biology.伽罗华和他的群论伽罗华是一位年轻的数学家， 1811年生於法国， 1830年当他18岁的时候，首次提出了“群”这个数学概念。
