群的阶与其元的阶之间的关系

石家庄铁道学院毕业论文

石家庄铁道学院毕业论文

群的阶与其元的阶之间的关系

The Relations between the Order of a Group

and the Orders of Its Elements

2008届 数理系 专 业 数学与应用数学 学 号 学生姓名

指导教师

完成日期 2008年５月20日

石家庄铁道学院毕业论文

摘 要

近世代数虽是一门较新的，较抽象的学科，但如今它已渗透到科学的各个领域，解决了许多著名的数学难题:像尺规作图不能问题，用根式解代数方程问题，编码问题等等．而群是近世代数里面最重要的内容之一，也是学好近世代数的关键．

本论文旨在从各个角度和方面来探讨群的阶与其元的阶之间的关系．具体地来说，本文先引入了群的概念，介绍了群及有关群的定义，然后着重讨论了有限群、无限群中关于元的阶的情况．并举了一些典型实例进行分析，之后又重点介绍了有限群中关于群的阶与其元的阶之间的关系的定理——拉格朗日定理，得出了一些比较好的结论．

在群论的众多分支中，有限群论无论从理论本身还是从实际应用来说，都占据着更为突出的地位．同时，它也是近年来研究最多、最活跃的一个数学分支．因此，在本文最后，我们介绍了著名的有限交换群的结构定理，并给出了实例分析．

关键词：群论 有限群 元的阶

石家庄铁道学院毕业论文

Abstract

The Modern Algebra is a relatively new and abstract subject, but now it has penetrated into all fields of science and solved a number of well-known mathematical problems, such as, the impossibility for Ruler Mapping problem, the solutions for algebraic equations with radical expressions, coding problems and so on. The group is one of the most important portions in the Modern Algebra, and also the key of learning it well. This paper aims at discussing the relations between the order of a group and the orders of its elements from all the angles and aspects. Specifically, this thesis firstly introduces the concept of a group and some relatives with it; secondly focuses on the orders of the elements in the finite group and the infinite group respectively, some typical examples are listed for analyses; thirdly stresses on the theorem - Lagrange's theorem on the relations between the order of a group and the orders of its elements in the finite group, accordingly obtaining some relatively good conclusion.

In the many branches of group theory, the finite group theory, whether from the theory itself or from the practical applications, occupies a more prominent position. At the same time, it is also one of the largest researches and the most active branches of mathematics in the recent years. Therefore, in this paper finally, we introduce the famous theorem of the structures on the finite exchanging groups, and give several examples for analyses.

Key words：group theory finite groups the orders of elements

石家庄铁道学院毕业论文

目 录

１ 绪 论 ······················································································································· 1 1.1 群论的概括 ················································································································· 1 1.2 群论的来源 ················································································································· 1 1.3 群论的思想 ················································································································· 2 2 预备知识 ························································································································· 2 2.1 群和子群 ····················································································································· 2

2.1.1 群的定义 ········································································································· 2 2.1.2 群的阶的定义 ································································································· 3 2.1.3 元的阶的定义 ································································································· 4 2.1.4 子群、子群的陪集 ························································································· 5 2.1.5 同构的定义 ····································································································· 6 2.2 不变子群与商群 ········································································································· 6

2.2.1 不变子群与商群 ····························································································· 6 2.2.2 Cayley(凯莱)定理 ························································································· 7 2.2.3 内直和和外直积的定义 ················································································· 8 3 群中元的阶的各种情况及其实例分析 ········································································· 8 3.1 有限群中关于元的阶 ································································································· 9

3.1.1 有限群中元的阶的有限性 ············································································· 9 3.1.2 有限群中关于元的阶及其个数的关系 ························································· 9 3.2 无限群中关于元的阶 ······························································································· 10

3.2.1 无限群G中，除去单位元外，每个元素的阶均无限 ······························ 10 3.2.2 无限群G中，每个元素的阶都有限 ·························································· 10 3.2.3 G为无限群，G中除单位元外，既有无限阶的元，又有有限阶的元 ··· 11 4 群的阶与其元的阶之间的关系 ··················································································· 11 4.1 拉格朗日(Lagrange)定理 ························································································· 11

4.1.1 拉格朗日定理 ······························································································· 11 4.1.2 相关结论 ······································································································· 12 4.2 有限交换群的结构定理 ··························································································· 13

4.2.1 有限交换群的结构定理 ··············································································· 13

石家庄铁道学院毕业论文

4.2.2 相关例子 ······································································································· 14 参 考 文 献 ······················································································································· 15 致 谢 ····························································································· 错误！未定义书签。

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