【作者】 李蕊；

【导师】 钱方生；

【作者基本信息】 哈尔滨师范大学 ， 基础数学， 2010， 硕士

【摘要】 自从伽罗华提出了置换子群的概念后,数学家们在它的基础上又提出了共轭置换子群、完全条件置换子群等一系列子群的定义.并且研究了它们对有限群的可解性等影响.之后,王燕鸣引入c-正规子群的概念,并研究了它对有限群结构的种种影响.还有一些工作者在此基础上又定义了弱c-正规子群、c-可补子群等等,研究它们的可解性、幂零性等. 2007年杨高才引入一个比c-正规子群更加广泛的概念-几乎正规子群,并研究具有几乎正规性质的有限群的结构.从而某些子群对有限群的结构有着重要的影响,本文就是通过对c-正规子群,几乎正规子群,共轭置换子群的研究来了解原群的结构.第一章为引言,主要介绍国内外群论的研究现状,以及本文研究的来源和主要研究结果.第二章为综述,主要介绍群论的发展及其内容,以及群论的研究方法.在第三章中,主要利用几乎正规子群的定义以及性质来研究Sylow子群的极大子群在几乎正规性的条件下对有限群结构的影响,并给出群G超可解的充分条件.在第四章中,本章主要是应用文献[6]中新定义的V (G),利用它的性质来研究c-正规子群的幂零性.在第五章中,本章主要是把群G的任何子群的Frattini子群为1的条件进行弱化为群G的任何真子群的Frattini子群为1,从而得到群G幂零与超可解的性质.更多还原

【Abstract】 Since the Galois proposed the concept of permutable subgroup, the mathemati-cians also put forward the concepts for conjugate Permutable subgroup, permutablesubgroup and a series of subgroups definitions which based on this concept. Theystudied these subgroups on the impact of solvability. Wang Y.M. proposed the con-cept of c-normal subgroups and studied its impact on finite group. Some workersbased on these results defined weak c-normal subgroups, c-complemented subgroupand study their solvability and so on. In 2007, Yang G.C. propose a almost normalsubgroup which broader than c-normal subgroups, and study the structure of finitegroups with satisfied the properties of almost normal subgroup.Certain subgroups have a significant impact on the structure of the finitegroups. This paper studies the structure of the original group through the c-normalsubgroup almost normal subgroup, conjugate-permutable subgroup.The first chapter is an introduction, introduces the domestic and abroad re-search on the situation of the group theory, as well as the source and the maincontents of this paper.The second chapter, introduces the development content and research methodsof group theory.In the third chapter, studies the impact of the maximal subgroup Sylow sub-groups on the finite groups under the formal condition by using the definition andnature of the almost normal subgroups and given the su?cient condition of super-solvable of super-G.In the fourth chapter, studies c-normal subgroup nilpotency by using the natureof a new definition of V(G) in references [6].In the fifth chapter, the author weakens the conditions of subgroup of Frattini,any subgroup of G, into that any real subgroup Frattini is 1, so the nature ofnilpotent and supersolvable of group G.更多还原