摘要
极大子群、极小子群和交换子群等是有限群中三类非常重要的子群,它们在有限群结构的研究中起着非常关键的作用。本文主要利用极大子群、极小子群和交换子群等的数量性质和正规性质对有限群的结构进行了研究,得到了一些新的结果。全文共分四章,主要内容如下:
第一章给出了全文所用到的符号和基本概念。
第二章主要利用极大子群共轭类型分别对一般的交错群、对称群、可解群、非可解群和含非交换单子群的有限群等进行了刻画,得到了若干新的结果。
第三章首先给出了一类非可解群,说明文献[3]定理2中的假设是完全必要的,进而分别给出例子对相关文献中在研究有限群的p-幂零性、可解性和超可解性时所给定假设的必要性进行了说明。其次,给出了文献[3]中两个主要定理的若干新的推广。最后,利用极大子群指数得到了关于可解群的一个新的结果。
第四章对交换子群个数给定的有限群进行了研究,给出了非平凡交换子群个数为5和6的有限群的完全分类,并且证明了非平凡交换子群个数小于12的有限群必定可解。
Maximal subgroup, minimal subgroup and abelian subgroup are three classes of very important subgroups, which played an important part in the study of the structure of finite groups. In the thesis, we study the structure of finite groups through considering the quantitative properties and normal properties of maximal subgroups, minimal subgroups and abelian subgroups, some new results are obtained. It consists of four chapters.
In Chapter 1, we introduce some symbols and basic concepts which are used in the thesis.
In Chapter 2, we mainly use the type of conjugacy classes of maximal subgroups to characterize the ordinary alternating groups, symmetric groups, solvable groups, non-solvable groups and finite groups which have a non-abelian simple subgroup, et al, some new results are obtained.
In Chapter 3, we firstly give a class of non-solvable groups, which show that the hypothesis for Theorem 2 in [3] is very essential. Moreover, we give some examples to show that some hypotheses for a finite group to be p-nilpotent, solvable and supersolvable in some papers can not be removed, respectively. Secondly, some new generalizations for two important theorems in [3] are obtained. Lastly, a new result for solvable groups is given by considering the indices of maximal subgroups.
In Chapter 4, we study finite groups whose the number of abelian subgroups are given. Then, we obtain a complete classification for finite groups which have exactly 5 or 6 non-trivial abelian subgroups, and we prove that finite groups which have less than 12 non-trivial abelian subgroups are solvable.
引文
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