群论中的几个公开问题研究
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摘要
本学位论文将着力于研究解决群论中的几个公开问题,内容主要包括两个方面:一是研究与群类理论相关的群的子群结构和群类的代数结构方面的几个问题;二是利用子群置换性质刻画有限群结构并解决了相关的一个公开问题.全文共分为七章.
第一章引言,主要介绍本论文的研究背景和取得的主要成果.
第二章用于介绍本文中一些常用的概念,符号及一些已知的基本结果.
第三章研究有限群的覆盖子群和内射子.本章解决了由L.A.Shemetkov院士提出的关于(?)-覆盖子群的存在性与共轭性的一个公开问题,还解决了郭文彬教授提出的关于(?)-内射子的存在性与共轭性的一个公开问题,并使得由俄罗斯科学院西伯利亚分院每年出版的《群论中未解决的问题—THE KOUROVKANOTEBOOK》中的一个公开问题(问题12.96)得到了进展.
第四章用于研究Fitting类的模律.在《群论中未解决的问题—THEKOUROVKA NOTEBOOK》中有下列公开问题(问题14.47):“有限可解群的Fitting类格为模格吗?”本章我们给出局部Fitting类的格为模格成立的一个条件,证明了:设(?):LR(x),(?)=LR(y)和(?)=LR(f)为局部Fitting类,x,y和f分别为它们的极小H-函数且x≤f.如果对所有的p∈P和x(p)≠φ,y(p)≠φ,有x(p)∨y(p)=S_n(G|G=G_(x(p))G_(y(p))),那么以下模律成立:((?)∨_l(?))∩(?)=(?)∨_l((?)∩(?)).
第五章研究Fitting类的Shemetkov问题.我们在π-可解群的范围内,对Fitting类上的Shemetkov问题给予了肯定的回答,证明了:对任何局部Fitting类(?),K_π((?))也是一个局部Fitting类.应用这一结论,我们得到了关于π-可解群的Hallπ-子群的(?)-根的一个刻画;而且对于一个H-函数f,我们也给出了π-可解群的f-根的一些描述.
第六章研究X-半置换子群.我们利用X-半置换子群给出了超可解群的一些新的特征定理.其中,我们证明了如果一个群G的所有2-极大子群在G中是F(G)-半置换的,则G是超可解的.这一定理解决了由郭文彬,A.N.Skiba和K.P Shum在《代数杂志》中提出的一个公开问题.当G的所有Sylow子群在G内是F(G)-半置换时,我们证明了不仅G是超可解的,而且对任何素数p,G/O_(p′,p)(G)是循环的.同时,我们也证明了一个群G是超可解的如果它的所有Sylow子群的每个极大子群在G内是F(G)-半置换的.在这一章,我们还将利用Sylow p-子群2-极大子群的F(G)-半置换性给出一个群的p-幂零性的一个刻画.
第七章我们给出了s-条件置换子群的概念,并利用这一概念给出了一个群属于给定群系的一些判别准则,同时也给出了p-超可解群的一些结构性定理.譬如,我们证明了:如果G为一个p-可解群.则G为p-超可解群的两个充要条件是:1)对于G的任一非Fractini p-主因子H/K,存在G的Sylow p-子群的某个不覆盖H/K的极大子群P_1在G中s-条件置换;2)存在G的正规子群N使得G/N为p-超可解群且N的Sylow p-子群的任一极大子群或者在G内有p-超可解群补充或者在G中s-条件置换.
The main object of this paper is to attack some open questions in the theory of groups.
In ChapⅠ,we introduce the background of the paper and some main results obtained in this paper.
In ChapⅡ,we introduce the notations and terminologies used in this paper and we list also some known results.
The main purpose in ChapⅢis to research covering subgroups and injectors. In this chapter,we solve an open question on(?)-covering subgroups proposed by L. A.Shemetkov and an open question on(?)-injectors proposed by Wenbin Guo.The results in this chapter also shed new light on a problem of Unsolved Problems in Group Theory-THE KOUROVKA NOTEBOOK(Problem 12.96).
We study the modular law on Fitting class in ChapⅣ.In Unsolved Problems in Group Theory-THE KOUROVKA NOTEBOOK,there is such an open problem (Problem 14.47):"Is the lattice of all soluble Fitting classes of finite groups modular?" Work on this Problem,in this chapter,we proved that:
Let(?)=LR(x),(?)=LR(y)and(?)=LR(f)be local Fitting classes with least H-function x,y and f respectively,and x≤f.If x and y satisfy that x(p)∨y(p)= S_n(G|G=G_(x(p))G_(y(p))for all p∈P where x(p)≠φand y(p)≠φ,then the following modular law holds:((?)∨_l(?))∩(?)=(?)∨_l((?)∩(?)).
In ChapⅤ,we research a Shemetkov's question on Fitting class.In the universe (?)_π(?)_(π＇)—the class of allπ-soluble group,we give a positive answer to this question. We prove that:
For any set of primesπ- and any local Fitting class(?),the Fitting class K_π((?))is a local Fitting class.
By using this result,we give some applications.In particular,the(?)-radical of a Hallπ-subgroup of a finite soluble group is described.
In ChapⅥ,we give some new characterizations of finite supersoluble groups by using the property of X-semipermutable subgroups.In particularly,we prove that if all 2-maximal subgroups of a group G is F(G)-semipermutable in G,then G is supersoluble. This theorem give a positive answer to an open question proposed by Wenbin Guo,A.N.Skiba and K.P.Shum on Journal of Algebra.Under the condition that all Sylow subgroups of G are F(G)-semipermutable in G,we obtain that G is supersoluble and,furthermore,the factor G/O_(p＇,p)(G)is cyclic for any prime p.Also,we prove that a group G is supersoluble if all maximal subgroups of its Sylow subgroups are F(G)-semipermutable in G.In this chapter,by using the F(G)-semipermutability of 2-maximal subgroups of some Sylow subgroups,we obtain a criterion on p-nilpotent groups.
In ChapⅦ,we give a definition on s-conditionally permutable subgroups.By this notation,we give some criterions of groups belonging to some given saturated formation. We also give some characterizations on p-supersoluble groups.For example, we prove:if G is a p-soluble group,then G is p-supersoluble if and only if one of the following statement hold:1)for any non-Frattini p-chief factor H/K of G,there exists a maximal subgroup P_1 of a Sylow p-subgroup of G such that P_1 is s-conditionally permutable in G and does not cover H/K.2)there exists a normal subgroup N of G such that G/N is p-supersoluble and any maximal subgroup of a Sylow p-subgroup of N either has a p-supersoluble supplement in G or is s-conditionally permutable in G.
第一章引言,主要介绍本论文的研究背景和取得的主要成果.
第二章用于介绍本文中一些常用的概念,符号及一些已知的基本结果.
第三章研究有限群的覆盖子群和内射子.本章解决了由L.A.Shemetkov院士提出的关于(?)-覆盖子群的存在性与共轭性的一个公开问题,还解决了郭文彬教授提出的关于(?)-内射子的存在性与共轭性的一个公开问题,并使得由俄罗斯科学院西伯利亚分院每年出版的《群论中未解决的问题—THE KOUROVKANOTEBOOK》中的一个公开问题(问题12.96)得到了进展.
第四章用于研究Fitting类的模律.在《群论中未解决的问题—THEKOUROVKA NOTEBOOK》中有下列公开问题(问题14.47):“有限可解群的Fitting类格为模格吗?”本章我们给出局部Fitting类的格为模格成立的一个条件,证明了:设(?):LR(x),(?)=LR(y)和(?)=LR(f)为局部Fitting类,x,y和f分别为它们的极小H-函数且x≤f.如果对所有的p∈P和x(p)≠φ,y(p)≠φ,有x(p)∨y(p)=S_n(G|G=G_(x(p))G_(y(p))),那么以下模律成立:((?)∨_l(?))∩(?)=(?)∨_l((?)∩(?)).
第五章研究Fitting类的Shemetkov问题.我们在π-可解群的范围内,对Fitting类上的Shemetkov问题给予了肯定的回答,证明了:对任何局部Fitting类(?),K_π((?))也是一个局部Fitting类.应用这一结论,我们得到了关于π-可解群的Hallπ-子群的(?)-根的一个刻画;而且对于一个H-函数f,我们也给出了π-可解群的f-根的一些描述.
第六章研究X-半置换子群.我们利用X-半置换子群给出了超可解群的一些新的特征定理.其中,我们证明了如果一个群G的所有2-极大子群在G中是F(G)-半置换的,则G是超可解的.这一定理解决了由郭文彬,A.N.Skiba和K.P Shum在《代数杂志》中提出的一个公开问题.当G的所有Sylow子群在G内是F(G)-半置换时,我们证明了不仅G是超可解的,而且对任何素数p,G/O_(p′,p)(G)是循环的.同时,我们也证明了一个群G是超可解的如果它的所有Sylow子群的每个极大子群在G内是F(G)-半置换的.在这一章,我们还将利用Sylow p-子群2-极大子群的F(G)-半置换性给出一个群的p-幂零性的一个刻画.
第七章我们给出了s-条件置换子群的概念,并利用这一概念给出了一个群属于给定群系的一些判别准则,同时也给出了p-超可解群的一些结构性定理.譬如,我们证明了:如果G为一个p-可解群.则G为p-超可解群的两个充要条件是:1)对于G的任一非Fractini p-主因子H/K,存在G的Sylow p-子群的某个不覆盖H/K的极大子群P_1在G中s-条件置换;2)存在G的正规子群N使得G/N为p-超可解群且N的Sylow p-子群的任一极大子群或者在G内有p-超可解群补充或者在G中s-条件置换.
The main object of this paper is to attack some open questions in the theory of groups.
In ChapⅠ,we introduce the background of the paper and some main results obtained in this paper.
In ChapⅡ,we introduce the notations and terminologies used in this paper and we list also some known results.
The main purpose in ChapⅢis to research covering subgroups and injectors. In this chapter,we solve an open question on(?)-covering subgroups proposed by L. A.Shemetkov and an open question on(?)-injectors proposed by Wenbin Guo.The results in this chapter also shed new light on a problem of Unsolved Problems in Group Theory-THE KOUROVKA NOTEBOOK(Problem 12.96).
We study the modular law on Fitting class in ChapⅣ.In Unsolved Problems in Group Theory-THE KOUROVKA NOTEBOOK,there is such an open problem (Problem 14.47):"Is the lattice of all soluble Fitting classes of finite groups modular?" Work on this Problem,in this chapter,we proved that:
Let(?)=LR(x),(?)=LR(y)and(?)=LR(f)be local Fitting classes with least H-function x,y and f respectively,and x≤f.If x and y satisfy that x(p)∨y(p)= S_n(G|G=G_(x(p))G_(y(p))for all p∈P where x(p)≠φand y(p)≠φ,then the following modular law holds:((?)∨_l(?))∩(?)=(?)∨_l((?)∩(?)).
In ChapⅤ,we research a Shemetkov's question on Fitting class.In the universe (?)_π(?)_(π＇)—the class of allπ-soluble group,we give a positive answer to this question. We prove that:
For any set of primesπ- and any local Fitting class(?),the Fitting class K_π((?))is a local Fitting class.
By using this result,we give some applications.In particular,the(?)-radical of a Hallπ-subgroup of a finite soluble group is described.
In ChapⅥ,we give some new characterizations of finite supersoluble groups by using the property of X-semipermutable subgroups.In particularly,we prove that if all 2-maximal subgroups of a group G is F(G)-semipermutable in G,then G is supersoluble. This theorem give a positive answer to an open question proposed by Wenbin Guo,A.N.Skiba and K.P.Shum on Journal of Algebra.Under the condition that all Sylow subgroups of G are F(G)-semipermutable in G,we obtain that G is supersoluble and,furthermore,the factor G/O_(p＇,p)(G)is cyclic for any prime p.Also,we prove that a group G is supersoluble if all maximal subgroups of its Sylow subgroups are F(G)-semipermutable in G.In this chapter,by using the F(G)-semipermutability of 2-maximal subgroups of some Sylow subgroups,we obtain a criterion on p-nilpotent groups.
In ChapⅦ,we give a definition on s-conditionally permutable subgroups.By this notation,we give some criterions of groups belonging to some given saturated formation. We also give some characterizations on p-supersoluble groups.For example, we prove:if G is a p-soluble group,then G is p-supersoluble if and only if one of the following statement hold:1)for any non-Frattini p-chief factor H/K of G,there exists a maximal subgroup P_1 of a Sylow p-subgroup of G such that P_1 is s-conditionally permutable in G and does not cover H/K.2)there exists a normal subgroup N of G such that G/N is p-supersoluble and any maximal subgroup of a Sylow p-subgroup of N either has a p-supersoluble supplement in G or is s-conditionally permutable in G.
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