子群的πF-可补充性、SΦ-嵌入性、CAP-拟正规性以及U-超中心性等研究

发布时间：2018-03-28 08:20

本文选题：有限群　切入点：饱和群系　出处：《中国科学技术大学》2017年博士论文

【摘要】：本学位论文主要研究子群的π(F)-可补充性、π(F)-嵌入性、SΦ-嵌入性以及CAP-拟正规性对有限群结构的影响,同时研究了正规子群的U-超中心性.本论文所涉及的群均是有限群.全文共分五章.第一章简单介绍了本论文的研究背景以及主要结果.第二章给出了本论文所需的一些概念和常用结论.第三章,在前人研究的基础上,我们给出了π(F)-可补充子群和π(F)-嵌入子群的新概念,并讨论它们对有限群结构的影响,得到了一个群是p-幂零的或者超可解的一些判别准则和充分必要条件.第四章进一步发展了对嵌入子群的研究,并讨论了 Frattini子群在子群嵌入性中的应用.我们建立了 CAP-拟正规子群和SΦ-嵌入子群的理论,并得到了一个群是p-幂零群,p-超可解群以及更一般地,属于某个包含所有超可解群的饱和群系的条件.这些结果统一并推广了大量已知的成果.第五章利用U-Φ-可补充子群和弱Π-嵌入子群研究了正规子群的U-超中心性.特别地,针对超可解群系U,我们给出了有限群G的一个正规子群包含在Z_U(G)或者Z_(pU)(G)中的条件.此外,我们还得到了一个群是p-幂零群或者p-超可解群的新的判别准则.
[Abstract]:In this dissertation, we mainly study the effects of 蟺 -F ~ + -complement, 蟺 -F _ (+) -embedding and CAP-quasi _ normality on the structure of finite groups, and the influence of S 桅 -embeddedness and CAP-quasi-normality on the structure of finite groups. At the same time, we study the U-hypercentricity of normal subgroups. The groups involved in this paper are all finite groups. The thesis is divided into five chapters. The first chapter briefly introduces the research background and main results of this thesis. Some necessary concepts and common conclusions. Chapter III, On the basis of previous studies, we give the new concepts of 蟺 ~ (+) F ~ (+) ~ (-) _ _ _. Some criteria and sufficient and necessary conditions for a group to be pnilpotent or supersolvable are obtained. In Chapter 4, the study of embedded subgroups is further developed. We also discuss the application of Frattini subgroups to the embedding of subgroups. We establish the theory of CAP-quasi normal subgroups and S 桅 -embedded subgroups, and obtain that a group is pnilpotent group p-supersolvable group and more generally, These results unify and generalize a large number of known results. In chapter 5, we study the U-hypercentricity of normal subgroups by using U- 桅 -complementary subgroups and weakly 蟺 -embedded subgroups. For supersolvable formations U, we give a condition for a normal subgroup of a finite group G to be contained in Z _ S _ U _ G) or Z _ S _ p _ U _ G). In addition, we obtain a new criterion that a group is a p-nilpotent group or a p-supersolvable group.
【学位授予单位】：中国科学技术大学
【学位级别】：博士
【学位授予年份】：2017
【分类号】：O152.1

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