抽象代数1

理解群和交换群的定义，群的一些简单的性质以及逆元和单位元在群中的作用。了解同群有密切关系但比群更广泛的代数系统半群。掌握群中元素的阶的概念和表示方法。会求一些简单群中的指定元素的阶。理解子群的概念和群的分类：平凡子群及真子群。知道给定群的子群的单位元和逆元与该群的关系。掌握非空子集做成子群的充要条件。知道中心元素的概念，会找一些简单群的中心。
通过以上知识的学习和习题的训练，培养学生的抽象思维能力和严密的逻辑推理能力，使学生们将受到良好的代数训练，并为进一步学习数学得到一个扎实的代数基础。
rse Description
Abstract Algebra Iis one of the important basic mathematics courses,whichintroducessome basic theories and such concepts asgroups, rings, fields (and modules).It aims at enabling students to: (1)understand the concept of “transformation”,and differentiate it from “mapping”; (2) know about algebra operations,determine whether a givenoperation isanalgebraone, and verify whether a given algebra operation satisfiesassociative, commutative and distributive laws;(3)determinewhethertwo different algebraic systems areisomorphic with the state; (4)understandthe relationship betweenequivalence relationsand collection classification, know how to identify a collection classification with a given equivalence relation, oridentify an equivalence relation with agiven collection classification; (5) understand the definitions of “group”and“exchange group”,some simple properties of a group,andthe roles that an inverse element and an identity played in a group respectively;(6) learn asemi-groupin analgebra systemwhich is closely related with a group but is more widely used;(7) grasp the concept and representation of “order” of elements in a group, andknow how toget the order of a given element insome simple groups; (8)understand the concept of “subgroup” and group classification: trivial subgroup and real subgroup, know the relation of an inverse element and an identity of a subgroup to the whole group, remember the necessary and sufficient conditions for being a nonempty subgroup; (9) know what is a central element and how tofindthe center of some simple groups; (10) understand thegeneration of a cyclic group and the relationship between a cyclic group and itssubgroup, and determine whether an element in a cyclic group withnorders can generate the cyclic group; (11) know what is a transformation group, and understand the linkage betweenpopulation and population changes; (12) understand the definitions of “permutation group”, “circulation”, and “change of circulation”, and know how to indicate the replacement cycle by using circulation and product; (13) understand the conceptsof“odd replacement”and“even replacement”andtheir relation, and mastersimple arithmeticoperations involved:multiplication in replacement, inversereplacement, and order of replacement; (14) understand the definitions of“coset”, “index”, andLagrange Theorem, and know about the relation of a coset and an index to the order of a group according toLagrange Theorem; (15) grasp the definition of “normal subgroup”, its simple properties, the conditions for being a normal subgroup, normal subgroups inhomomorphism, and the multiplication of normal subgroups; (16) understand what is a factor group and know about one of its applications; (17) know what is a ring, a commutative ring, and a non- commutative ring respectively, and identify whether an algebraic systemon agivencollection constitutes a ring; (18) understand the concept of “cyclic ring”.
Byintroducingthe above knowledge and correspondingexercises,the course is intended to foster students’ abstract thinkingandlogical reasoning so that they willbe well-trainedinalgebra, and help to establisha solid foundation fortheir further mathematic study.
抽象代数1专业主干课程简介
(Brief Introduction to the Major Courses ofAbstract Algebra I)
课程名称
抽象代数1
学分(Credits)
3
Course Name
Abstract Algebra I
学时(Hours)
54
课程简介（300字以内）
《抽象代数I》是数学学科的重要基础课程之一，主要介绍群，环，域（以及模）的基本概念和基本理论。让学生了解变换的概念,区分变换与映射的不同。理解代数运算的概念,会判断给定的运算是否代数运算。对于给定的代数运算，会验证是否满足结合律，交换律以及左右分配律。给定两个不同的代数系统，会判断二者是否同态或者同构。最后，在这一部分还要求理解等价关系和集合分类之间的关系，对给定的等价关系，如何确定一个集合的分类，反之，给定一个集合的分类又掌握确定怎样的一个等价关系的方法。
理解循环群的生成，循环群的子群和循环群的关系。会判断n阶循环群中的一个元素是否可以生成这个循环群。了解变换群的概念，理解抽象群和变化群之间的联系。理解置换群，循环和对换的定义，会用循环和循环的乘积来表示置换。了解奇置换和偶置换的概念和它们之间的关系。掌握置换的简单运算：置换间的相乘，置换逆的求法和置换的阶。理解陪集，指数的定义和Lagrange定理的内容。了解Lagrange定理所给出的陪集和指数与群的阶之间的关系。掌握正规子群的定义和简单性质，子群做成正规子群的条件。在同态映射下的正规子群以及正规子群相乘的状况。了解商群及商群的一个应用。理解环的定义，以及交换环和非交换环的概念。会判断给定的一个集合上的运算的代数系统是否构成环。了解循环的概念。