子群的正规性和正规化子条件对群结构的影响
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摘要
长期以来,利用子群的某种正规性来研究有限群的结构一直都是有限群理论研究的重要课题之一.特别是有限单群分类完成之后,群论学者们定义了一些新的广义正规性来刻画有限群的结构,得到了相当丰硕的成果,这给有限群的研究和发展提供了强有力的推动作用.正规子群是各种广义正规性中最基本和最核心的概念,子群的正规性在有限群的研究中起着十分重要的作用.本文的前几部分推广了子群的正规性得到了一些较弱的正规性和给出了与共轭置换性对偶的嵌入性质,自共轭置换子群的新概念,并利用这些置换性和嵌入子群来研究有限群的结构.安排如下：
第一章,我们主要介绍本文研究要用到的一些概念与周知的结论.
第二章,我们介绍了一种新的嵌入子群的定义,自共轭置换子群和利用自共轭置换性研究了可解丁-群,极小非丁-群的新特征和给出了偶阶极大子群是PSC-群的结构描述.特别是根据施武杰教授给出的一般线性群的子群的结构,我们给出了一些特殊单群和非可解群的结构刻画.这章的部分结果主要发表在Communication in Algebra, Joural of Algebra and its Application和Acta Mathematica Sinica, English Series.
第三章,结合子群的置换性和子群的可补性,我们引进了SNS-置换子群,SS-拟正规嵌入子群和弱SS-拟正规嵌入子群的概念.进而讨论极大子群和同阶子群集合的元素满足SNS-置换性,SS-拟正规嵌入子群假设条件下的有限群的结构,并把相应结果推广到群系框架.这章的部分结果主要发表在Publication Mathematic Debrecen.
第四章,研究Sylow子群P的秩d个(也是最小生成元个)极大子群的集合,我们定义为集合Md(P),满足拟正规嵌入子群,H-子群和SS-拟正规性条件下有限群的结构.我们得到了p-幂零群,超可解群的一些新的判别准则.并且我们把相应结果推广到群系框架.改进,统一和推广了最近的一些结果.特别是解决了李世荣教授提出的一个问题和改进了段学复院士,陈重穆教授,张继平教授和张来武教授在1984年在北京大学国际群论研讨会上提出的减少Sylow子群P极大子群个数对群结构影响的结果.在1984年,他们就提出了减少Sylow子群P的极大子群的个数集合的思想.他们首先考虑到的也是从最小生成元个数d出发来给出这样的集合,为了方便我们定义这个集合DCZZ(P),并且这个集合的个数是f(d)=(pd-2-1)/(p-1),在这里d>2是P的最小生成元个数(事实上,Sylow子群所有极大子群的个数是(pd-1)/(p-1)).当d比较大时,我们用极限的法则可以判定f(d)>>d.并且我们也给出了比d小的极大子群的集合就不能给出p-幂零群的判别准则.这章的部分结果主要发表在Journal of Group Theory, Fronter Mathematic in China和Algebra Collq.
作为正规性概念的对偶思想.1934年,Baer引入了Norm的概念.给定群G,G的Norm记作N(G)被定义为G的所有子群的正规化子的交.
第五章,我们推广了Norm的概念,考虑非循环子群,子群的导子群和子群的幂零剩余的正规化子的交,即也称作非循环子群,子群的导子群和子群的幂零剩余的Norm特别,推广了著名的Ito和Burnside关于p-幂零的准则,给出了p-可解群的p-长的刻画.最后我们研究了非循环子群,子群的导子群和子群的幂零剩余的Norm得到了一些无限群的初等结果.首次研究了Norm对无限群结构的影响.这章的部分结果主要发表在Journal of Algebra和Journal of Korea Mathematic Soc.
It has been one of the important topics to study the structure of the finite groups using the normality of subgroups for a long time. In particular, after finite simple groups were classified. Not only many new concepts have been introduced but also fruitful results have been ob-tained, which pushed forward the study and the development of finite groups theory. The concept of normal subgroup is most fundamental and the normalities of subgroups are very important in the research of finite groups. In the first few parts of this thesis, we generalize the normalities of subgroups and then get some weaker normalities of sub-groups, give the dual concept of conjugate permutable, self-conjugate permutable and the dual of normality, the normalizer condition. And we study the structure of finite groups by using these weaker normali-ties of subgroups, embedded subgroups and the normalizer condition. It consists of five chapters.
Chapter 1. We introduce some notations and list some known re-sults needed in the paper.
Chapter 2. We introduce a new embedded subgroups concept, self-conjugate permutable subgroup and we investigate new characterations of solvabel T-groups, minimal non-T-groups and give the structure of groups whose maximal subgroups of even order are PSC-groups by con-sidering self-conjugate permutable subgroup. In particular, we give new characteration of some special simple groups and non-solvable groups by considering self-conjugate permutable subgroup and by Wujie Shi's results about the structure of subgroups of special liner groups. A part content of this chapter will be published in Communication in Algebra, Joural of Algebra and its application and Acta Mathematica Sinica, En-glish Series.
Chapter 3. We define the SNS-permutable subgroups, SS-quasinormally embedded subgroups and weakly SS-quasinormally embedded subgroups by permutablity and complement. Furthemore, we investigate the struc-ture of the finite groups whose the maximal subgroups and the elments of the set of subgroups with the same order satisfy the SNS-permutability, SS-quasinormally embedded subgroups and extend related results to for-mation. A part content of this chapter will be published in Publpcation. Mathematics. Debrecen.
Chapter 4. We investigate the influence of the quasinormally embed-ded subgroups,H-subgroups, SS-quasinormality of maximal subgroups of Sylow subgroup P with rank d (i.e., the smallest generator number) on the structure of finite groups and obtain some new criterions for p-nilpotency and supersolubility of finite groups. The obtained results improve, unify and generalize some recent related results. In particular, we solve a open problem of Li Shirong and improve Duan Xuefu, Chen Zhongmu, Zhang Jiping and Zhang Laiwu giving the idea of reducing the maximal set of Sylow subgroup. In 1984, they gave the idea of reducing the maximal set of Sylow subgroup. They also considered the smellest generator number d and gave DCZZ(P) is a set of maximal subgroups of Sylow subgroup P such that|DCZZ(P)|= f(d)= (pd-2-1)/(p-1) (In fact, the number of all maximal subgroups of Sylow subgroup P is (pd-1)/(p-1)). Moreover, we have|f(d)|》d when d→∞.At the same time, we give a example such that the smallest generator number d is the best possible. A part content of this chapter will be published in Journal of Group Theory, Fronter Mathematic in China and Algebra Collq.
For the dual of normality. In 1934, Baer define the concept of Norm. Let G be a group and the Norm of G by N(G)= (?)H≤GNG(H).
Chapter 5. We generalize the concept of Norm and give the in- tersection of normalizer of non-cyclic subgroups, the dirived subgroups of subgroups and the nilpotent residual of subgroups, i.e., the Norm of non-cyclic subgroups, the dirived subgroups of subgroups and the nilpo-tent residual of subgroups. In particular, we generalize the p-nilpotent criterion of Ito and Burnside and give the p-lentgh of p-solvable. At last, we study the Norm of non-cyclic subgroups, the dirived subgroups of subgroups and the nilpotent residual of subgroups and obtain some basic results of infinite groups. We first investigate the structure of the infinite groups by Norm. A part content of this chapter will be published in Journal of Algebra and Journal of Keroa Mathematic Soc.
第一章,我们主要介绍本文研究要用到的一些概念与周知的结论.
第二章,我们介绍了一种新的嵌入子群的定义,自共轭置换子群和利用自共轭置换性研究了可解丁-群,极小非丁-群的新特征和给出了偶阶极大子群是PSC-群的结构描述.特别是根据施武杰教授给出的一般线性群的子群的结构,我们给出了一些特殊单群和非可解群的结构刻画.这章的部分结果主要发表在Communication in Algebra, Joural of Algebra and its Application和Acta Mathematica Sinica, English Series.
第三章,结合子群的置换性和子群的可补性,我们引进了SNS-置换子群,SS-拟正规嵌入子群和弱SS-拟正规嵌入子群的概念.进而讨论极大子群和同阶子群集合的元素满足SNS-置换性,SS-拟正规嵌入子群假设条件下的有限群的结构,并把相应结果推广到群系框架.这章的部分结果主要发表在Publication Mathematic Debrecen.
第四章,研究Sylow子群P的秩d个(也是最小生成元个)极大子群的集合,我们定义为集合Md(P),满足拟正规嵌入子群,H-子群和SS-拟正规性条件下有限群的结构.我们得到了p-幂零群,超可解群的一些新的判别准则.并且我们把相应结果推广到群系框架.改进,统一和推广了最近的一些结果.特别是解决了李世荣教授提出的一个问题和改进了段学复院士,陈重穆教授,张继平教授和张来武教授在1984年在北京大学国际群论研讨会上提出的减少Sylow子群P极大子群个数对群结构影响的结果.在1984年,他们就提出了减少Sylow子群P的极大子群的个数集合的思想.他们首先考虑到的也是从最小生成元个数d出发来给出这样的集合,为了方便我们定义这个集合DCZZ(P),并且这个集合的个数是f(d)=(pd-2-1)/(p-1),在这里d>2是P的最小生成元个数(事实上,Sylow子群所有极大子群的个数是(pd-1)/(p-1)).当d比较大时,我们用极限的法则可以判定f(d)>>d.并且我们也给出了比d小的极大子群的集合就不能给出p-幂零群的判别准则.这章的部分结果主要发表在Journal of Group Theory, Fronter Mathematic in China和Algebra Collq.
作为正规性概念的对偶思想.1934年,Baer引入了Norm的概念.给定群G,G的Norm记作N(G)被定义为G的所有子群的正规化子的交.
第五章,我们推广了Norm的概念,考虑非循环子群,子群的导子群和子群的幂零剩余的正规化子的交,即也称作非循环子群,子群的导子群和子群的幂零剩余的Norm特别,推广了著名的Ito和Burnside关于p-幂零的准则,给出了p-可解群的p-长的刻画.最后我们研究了非循环子群,子群的导子群和子群的幂零剩余的Norm得到了一些无限群的初等结果.首次研究了Norm对无限群结构的影响.这章的部分结果主要发表在Journal of Algebra和Journal of Korea Mathematic Soc.
It has been one of the important topics to study the structure of the finite groups using the normality of subgroups for a long time. In particular, after finite simple groups were classified. Not only many new concepts have been introduced but also fruitful results have been ob-tained, which pushed forward the study and the development of finite groups theory. The concept of normal subgroup is most fundamental and the normalities of subgroups are very important in the research of finite groups. In the first few parts of this thesis, we generalize the normalities of subgroups and then get some weaker normalities of sub-groups, give the dual concept of conjugate permutable, self-conjugate permutable and the dual of normality, the normalizer condition. And we study the structure of finite groups by using these weaker normali-ties of subgroups, embedded subgroups and the normalizer condition. It consists of five chapters.
Chapter 1. We introduce some notations and list some known re-sults needed in the paper.
Chapter 2. We introduce a new embedded subgroups concept, self-conjugate permutable subgroup and we investigate new characterations of solvabel T-groups, minimal non-T-groups and give the structure of groups whose maximal subgroups of even order are PSC-groups by con-sidering self-conjugate permutable subgroup. In particular, we give new characteration of some special simple groups and non-solvable groups by considering self-conjugate permutable subgroup and by Wujie Shi's results about the structure of subgroups of special liner groups. A part content of this chapter will be published in Communication in Algebra, Joural of Algebra and its application and Acta Mathematica Sinica, En-glish Series.
Chapter 3. We define the SNS-permutable subgroups, SS-quasinormally embedded subgroups and weakly SS-quasinormally embedded subgroups by permutablity and complement. Furthemore, we investigate the struc-ture of the finite groups whose the maximal subgroups and the elments of the set of subgroups with the same order satisfy the SNS-permutability, SS-quasinormally embedded subgroups and extend related results to for-mation. A part content of this chapter will be published in Publpcation. Mathematics. Debrecen.
Chapter 4. We investigate the influence of the quasinormally embed-ded subgroups,H-subgroups, SS-quasinormality of maximal subgroups of Sylow subgroup P with rank d (i.e., the smallest generator number) on the structure of finite groups and obtain some new criterions for p-nilpotency and supersolubility of finite groups. The obtained results improve, unify and generalize some recent related results. In particular, we solve a open problem of Li Shirong and improve Duan Xuefu, Chen Zhongmu, Zhang Jiping and Zhang Laiwu giving the idea of reducing the maximal set of Sylow subgroup. In 1984, they gave the idea of reducing the maximal set of Sylow subgroup. They also considered the smellest generator number d and gave DCZZ(P) is a set of maximal subgroups of Sylow subgroup P such that|DCZZ(P)|= f(d)= (pd-2-1)/(p-1) (In fact, the number of all maximal subgroups of Sylow subgroup P is (pd-1)/(p-1)). Moreover, we have|f(d)|》d when d→∞.At the same time, we give a example such that the smallest generator number d is the best possible. A part content of this chapter will be published in Journal of Group Theory, Fronter Mathematic in China and Algebra Collq.
For the dual of normality. In 1934, Baer define the concept of Norm. Let G be a group and the Norm of G by N(G)= (?)H≤GNG(H).
Chapter 5. We generalize the concept of Norm and give the in- tersection of normalizer of non-cyclic subgroups, the dirived subgroups of subgroups and the nilpotent residual of subgroups, i.e., the Norm of non-cyclic subgroups, the dirived subgroups of subgroups and the nilpo-tent residual of subgroups. In particular, we generalize the p-nilpotent criterion of Ito and Burnside and give the p-lentgh of p-solvable. At last, we study the Norm of non-cyclic subgroups, the dirived subgroups of subgroups and the nilpotent residual of subgroups and obtain some basic results of infinite groups. We first investigate the structure of the infinite groups by Norm. A part content of this chapter will be published in Journal of Algebra and Journal of Keroa Mathematic Soc.
引文
[1]R.K. Agrawal. Finite groups whose subnormal subgroups per-mutable with all Sylow subgroups. Proc.Amer.Soc.,47(1975):77-83.
[2]M.J. Alejandre, A. Ballester-Bolinches and M.C. Pedraza-Aguilera. Finite soluble groups with permutable subnormal subgroups. J. Algebra,240(2001):705-722.
[3]Z. Arad, M.B. Ward. New criteria for the solvability of finite groups. J.Algebra,77(1982):234-246.
[4]M. Asaad. On the supersolvability of finite groups. Annales Univ. Sci.Budapest,ⅫⅠ,18(1975):3-7.
[5]M. Asaad, p-Nilpotence and Supersolvable of Finite Groups, Com-munications in Algebra,34(2006):189-195.
[6]M. Asaad and A.A.Heliel.On S-quasinormal embedded subgroups of finite groups.J. Pure Appl. Algebra,165(2001):129-135.
[7]M. Assad, M. Ramadan and A. Shaalan. Infuence of S-quasinormality on maximal subgroups of Sylow subgroups of Fitting subgroups of a finite group. Arch.Math.,1991,56:521-527.
[8]R. Baer. Supersoluble immersion. Cana. J. Math.,11(1959):353-369.
[9]R. Baer, Der Kern eine charakteristische Untergruppe. Compositio Math.1(1934):254-283.
[10]R. Baer, Norm and hypernorm, Publ. Math. Debrecen,4(1956), 347-356.
[11]R. Baer, Classes of finite groups and their properties, Illinois J. Math,1 (1957),115-187.
[12]R. Baer, Zentrum und Kern von Gruppen mit Elementen un-endlicher Ordnung, Compositio Math.2 (1935) 247-249.
[13]A. Ballester-Boliches. Permutably embedded subgroups of finite sol-uble groups. Arch. Math.,65(1995):1-7.
[14]A. Ballester-Bolinches and M.C. Pedraza-Aquilera. Sufficient condi-tions for supersolvability of finite groups. J. Pure Appl.Algebra, 127(1998):223-228.
[15]A. Ballester-Bolinehes, Y. Wang and X. Guo. c-Supplemented sub-groups of finite groups, Glasgow Math. J.,42(2000):383-389.
[16]A. Ballester-Bolinches, R. Esteban-Romero, On finite T-groups. J. Aust. Math. Soc.,75(2003):181-191.
[17]S. Bauman and G. Ebter. A note on subnormal and abnormal chains. J. Algebra,36(1975):287-293.
[18]B. Baumeister. A characterization of the finite soluble groups. Arch. Math.,72(1999):167-176.
[19]J.C. Beidleman, B. Brewster and D. J.S. Robinson. Criteria for per-mutability to be transitive in finite groups. J. Algebra,222(1999): 400-412.
[20]J.C. Beidleman and H. Heineken. Finite soluble groups whose sub-normal subgroups permute with certain class of subgroups. J. Group Theorey,6(2003):139-158.
[21]J.C. Beidleman, H. Heineken, M. Newell. Center and norm, Bull. Austral. Math. Soc.69(2004) 457-464.
[22]V.A. Belonogov. Finite groups with three classes of maximal sub-groups, Math.Sb.,131(173)(1986),225-239.
[23]Y. Berkovich, Z. Janko. Structure of finite ρ-groups with given sub-groups. Ischia group theory 2004,13-93, Contemp. Math.,402.
[24]Ja.G., Berkovic. The existence of subgroups of a finite non-solvable group, Dokl. Akad. Nauk 156(1964):1255-1257.
[25]P. Bhattacharya, N.P. Mukherjee. On the intersection of a class of maximal subgroups of a finite group, J. Pure and Applied Algebra, 42(1986),117-124.
[26]Bianchi, M., Mauri, A.G.B., Herzog, M., Verardi, L., On finite solv-able groups in which normality is a transitive relation. J. Group Theory 3(2000):147-156.
[27]Z. Bozikov and Z. Janko, A complete classification of finite p-groups all of whose noncyclic subgroups are normal, Glasnik Matematicki. 44(2009),177-185.
[28]R. Brandl, Groups with few non-normal subgroups, Comm. Algebra, 23(1995),2091-2098.
[29]K. Brown, Finite groups with nilpotent maximal A-invariant sub-groups, Arch. Math. (Basel),22(1971),146-149.
[30]R.A. Bryce and John Cossey, The Wielandt subgroup of a finite soluble group. J. London Math. Soc.40(1989),244-256.
[31]J. Buckley. Finite groups whose minimal subgroups are normal. Math.Z.,116(1967),15-17.
[32]A.R. Camina, The Wielandt length of finite groups. J. Algebra 15(1970),142-148.
[33]R.W. Carter. Nilpotent self-normalizing subgroups of suluble groups. Math. Z.,75(1961):136-139.
[34]Guiyun Chen, On the structure of Frobenius groups and 2-Frobenius groups, J. Southwest China Normal University,20:5 (1995),485487. (Chinese)
[35]Guiyun Chen, On Thompson's conjecture, J. Algebra,185(1996), 184-193.
[36]Guiyun Chen, A new characterization of PS2(q), Southeast Asian Bull. of Math.22(1998),257-263.
[37]C. Chen. Generalization of Schur-Zassenhus Theorem. J. Math. 18(1998):290-294.
[38]陈重穆,内外∑-群与极小非∑-群,重庆：西南师范大学出版社,1988.
[39]陈重穆,Sriniasan定理的一个推广.西南师范大学学报(自然科学版),1(1987)：1-4.
[40]J.H. Conway, et al, Atlas of Finite Groups, Oxford: Clarendon Press,1985.
[41]Piroska Csorgo, Marcel Herzog, On Supersolvable Groups and the Nilpotator. Comm. Algebra,32(2004):609-620.
[42]W.E.Deskins, A note on the index complex of a maximal subgroup, Arch Math.(Basel),54(1990),236-240.
[43]R. Doerk and T. Hawkes. Finite soluble groups. Berlin-New York, Walter De Gruyter,1992.
[44]Xuefu Duan, Group Theory, Proceedings of an International Sym-posium held in Beijing, Aug.27-Sep.8,1984.
[45]L.M. Ezquerro. A contribution to the theory of finite supersolv-able groups. Rendieonti del Seminario Matematico della Universitdi Padova,1993,89:161-170.
[46]樊恽,李伟,能表示成三个真子群的并集的群,华中师范大学学报(自然科学版),1(2008)：1-3.
[47]樊恽,郭秀云,岑嘉评.关于子群的两种广义正规性的注记.数学年刊,27(2006)：169-176.
[48]A. Fattahi. Groups with only normal and abnormal subgroups. J. Algebra,28(1974):15-19.
[49]W. Feit, J.G. Thompson, Solvability of groups of odd order, Pacific J. Math.,13 (1963),775-1029.
[50]Flores, Ramon J., Foote, Richard M., Strongly closed subgroups of finite groups, Advances in Mathematics 222(2009):453-484.
[51]T. Foguel. Conjugate-permutable subgroups. J. Algebra,191(1997): 235-239.
[52]W. Gaschiitz. Praefrattini gruppen. Arch. Math.,13(1962):418-426.
[53]W. Gaschiitz. Gruppen,in dennen das Normalteilersein transitiv ist. J. Reine Angew. Math.,198(1957):87-92.
[54]J.D. Gillam. Cover-avoid subgroups in finite solvable groups. J.Algebra,29(1974):324-329.
[55]G. Glauberman, On Burnside's paqb theorem. Pacific J. Math. 56(1975):469-476.
[56]D. Goldschmidt,2-fusion in finite groups. Ann. of Math.99(1974): 70-117.
[57]D. Goldschmidt, Strongly closed 2-subgroups of finite groups. Ann. of Math.102(1975):475-489.
[58]D. Gorenstein. Finite Groups. New York:Harper Row,1968.
[59]D. Gorenstein. Finite Simple Groups. New York: Plenum Press,1982.
[60]W. Guo, K.P. Shum and A. Skiba. G-covering subgroup sys-tems for the classes of supersolvable and nilpotent groups. Israel J.Math.,2003,183:125-138.
[61]W. Guo. On F-supplemented subgroups of finite groups. Manuscripta Math.,2008,127:139-150.
[62]W. Guo, K.P. Shum and A.N. Skiba. X-permutable maximal sub-groups of Sylow subgroups of finite groups. Ukrainian Mathematical Journal,2006,5s(10):1471-1480.
[63]W. Guo, Shum K. P., and Skiba A. N., X-quasinormal subgroups, Siberian Math. J.48(2007):593-605.
[64]郭秀云,非幂零极大子群指数为素数幂的有限群,数学年刊(A辑),6(1994),721-725.
[65]X. Guo, P. Guo, K.P. Shum, On semi cover-avoiding subgroups of finite groups, J. Pure Appl. Algebra,209(2007),151-158.
[66]X. Guo, S. Li, P. Flavell, Finite groups whose abelian subgroups are TI-subgroups, J. Algebra,307(2007),565-569.
[67]郭秀云,岑嘉评,有限群的极小子群与p-幂零性,中国科学(A辑),9(2002),782-790.
[68]X. Guo and K.P. Shum. Cover-avoidance properties and the struc-ture of finite groups. J. Pure Appl.Algebra,2003.181:297-308.
[69]R.M. Guralnick, Subgroups of prime-power index in a simple group, J. Algebra,81 (1983),304-311.
[70]M. Hall. The theory of group. New York:Chelesea Pub.Corn.,1976.
[71]P. Hall. A characteristic property of soluble groups. J.London Math. Soc.,12(1937):198-200.
[72]M. Herzog, On finite simple groups of order divisible by three primes only, J. Algebra,10(1968),383-388.
[73]B. Huppert. Endliche gruppen I.Berlin/ Heidelberg/ New York:Springer-Verlag,1967.
[74]B. Huppert and N. Blackburn. Finite Groups Ⅲ.Berlin/ New York:Springer-Verlag,1982.
[75]I.M. Isaacs, Character theory of finite groups, New York-London, 1976.
[76]N. Ito. Note on (LM)-groups of finite orders.Kodai Math.Seminar report,3(1951):1-6.
[77]Z. Janko, Endliche Gruppen mit lauter nilpotenten zweitmaximalen untergrupprn, Math. Zeitschr.79(1962):422-424.
[78]P. Jin, Automorphisms of groups, Journal of Algebra,312(2007): 562-569.
[79]O.H. Kegel. Sylow gruppen and subnormalteiler endlicher grup-pen.Math.Z.,78(1962):205-221.
[80]O.H. Kegel, B.A.F. Wehrfritz, Locally Finite Groups, North-Holland, Amsterdam 1973.
[81]E.I., Khukhro, V.D., Mazurov, Finite groups with an automorphism of prime order whose centralizer has small rank. Journal of Algebra, 301 (2006):474-492.
[82]H. Kurzweil, B. Stellmacher, The Theory of Finite Groups:An Introduction, Springer-Verlag, New York,2004.
[83]R. Lauer. Dualization for saturation for locally defined formation. J.Algebra,52(1978):347-353.
[84]J.C. Lennox and S.E.Stonehewer, Subnormal subgroups of groups, Oxford,1987.
[85]S. Li, On minimal non-PE-group. J. Pure Appl. Algebra,132 (1998):149-158.
[86]S. Li, On minimal subgroups of finite groups, Comm.Algebra, 22(1994):1913-1918.
[87]S. Li. Q. Zhang. A note on p-supersoluble groups. Acta Mathematica Hungar.,116(2007):27-34.
[88]李世荣,非正规极大子群同阶类类数等于2的有限群,数学学报,3(1990),388-392.
[89]S. Li and X. He. On normally embedded subgroups of prime power order in finite groups. Comm. Algebra,2008,36(6):2333-2340.
[90]S. Li and Y. Zhao, Some finite nonsolvable groups characterized by their solvable subgroups, Acta Math. Sinica, New Series,4(1988): 5-13.
[91]X. Li, A characterization of the finite simple groups, J. Algebra, 245(2001),620-649.
[92]黎先华,有限单群的一个刻画与极大子群的指数集合,中国科学(A辑),1(2004),48-60.
[93]Y. Li and Y. Wang. On S-pquasinormally embedded subgroups of finite groups, J.Algebra,281(2004):109-123.
[94]Y. Li and Y. Wang. The influence of minimal subgroups on the structure of a finite group. Proc. Amer. Math.Soc.,2003,131(2):337-341.
[95]Y. Li, S. Qiao and Y. Wang. On weakly s-permutably embedded subgroups of finite groups. Comm. Algebra,2009,37:1086-1097.
[96]G.A. Miller and H.C. Moreno. Non-abelian groups in which every subgroup is abelian. Trans. Amer. Math. Soc.,4(1903):398-404.
[97]M. Miyamoto, Solvability of some groups, Hokkaido Math.J. 11(1982),106-110.
[98]H. Mousavi, On Finite groups with few non-normal subgroups, 27(1999),3143-3151.
[99]0. Ore. Contributions to the theory of groups of finite order. Duke Math. J.,5(1939):431-460.
[100]D.S. Passman, Nonnormal subgroups of p-groups, J.Algebra 15(1970),352-370.
[101]T.A. Peng, Finite groups with pronormal subgroups, Proc. Amer. Math. Soc.,20(1969):232-234.
[102]C.E. Praeger, Wujie Shi, A characterization of some alternating and symmetric groups, Comm. Algebra,5 (1994),1507-1530.
[103]G. Qian, F. Tang, Finite groups all of whose abelian subgroups are QTI-subgroups, J. Algebra,320 (2008),3605-3611.
[104]G. Qian, F. Tang, Finite groups all of whose subgroups are abelian or normal. Journal of Mathematical Research and Expo-sition,28(2008):273-278.
[105]G. Qian, Finite groups with n-maximal subgroup, Comm. Alge-bra, (in press)
[106]Ren Yongcai, Notes on S-quasinormal subgroups in finite groups, Proc. Amer. Math. Soc.,117(3)1993,631-636.
[107]D.J.S. Robinson. A note on finite groups in which normality is transitive. Proc. Amer. Math.Soc.,1968,19(4):933-937.
[108]D.J.S. Robinson. Groups which are minimal with respect to nor-mality being transitive. Pacific Journal of Math.31(1969):777-785.
[109]D.J.S. Robinson. A Course in the Theory of Groups (Second Edi-tion), Springer-Verlag, New York,1996.
[110]J.S. Rose. On finite insoluble groups with nilpotent maximal sub-groups, J. Algebra,48 (1977),182-196.
[111]N. Sastry. On minimal non-PN-groups. J. Algebra,65(1982):104-109.
[112]P. Schmid. Subgroups permutable with all Sylow subgroups. J. Algebra,1998,207:285-293.
[113]E. Schenkman. On the norm of a group, Illinois J. Math.4 (1960) 150-152.
[114]Shirong Li, Zhencai Shen, et. The influence of SS-quasinormality of some subgroups on the structure of finite groups. J. Algebra, 2008,319:4275-4287.
[115]Zhencai Shen, Wujie Shi, Finite groups with normally embedded subgroups, Joural of Group Theory 13(2010):257-265.
[116]Zhencai Shen, Wujie Shi, Qingliang Zhang. S-quasinormality of finite groups. Front. Math. (China), (5)(2010):329-339.
[117]Zhencai Shen, Shirong Li, Wujie Shi, Finite groups all of whose second maximal subgroups are PSC-groups. Journal of Algebra and Its Application.8(2009),229-242.
[118]Zhencai Shen, Shirong Li. On SS-quasinormal subgroups of finite groups. Communication In Algebra.36(2008):4436-4447.
[119]Shirong Li, Zhencai Shen. On the intersection of the normalizers of derived subgroups of all subgroups of a finite group, Journal of Algebra 323(2010),1349-1357.
[120]Zhencai Shen, Wujie Shi, Finite Groups with pronormal subgroups. Acta Math. Sinica, New Series (2011).
[121]Zhencai Shen, Wujie Shi, On SNS-permutable subgroups and the structure of finite groups. Publ. Math. Debrecen.78(2011):159-168.
[122]Zhencai Shen, Ni Du, Finite groups with H-subgroups, Algebra Collaq. (China),2012.
[123]施武杰,极大子群同阶类类数不大于2的有限群,数学年刊(A辑),5(1989),532-537.
[124]施武杰,关于单K4-群,科学通报,17(1991),1281-1283.
[125]Wujie Shi, A note on p-nilpotence of finite groups, J. Algebra, 241(2001):435-436.
[126]施武杰,毕建行,交错群的一个新刻画,科学通报,9(1989)：715-716.
[127]苏向盈.有限群的半正规子群.数学杂志,8(1988)：5-10.
[128]A.N. Skiba. On weakly s-permutable subgroups of finite groups. J.Algebra,315(2007):192-209.
[129]S. Srinivasan.Two sufficient conditions for supersolvability of finite groups. Israel J. Math.,35(1980):210-214.
[130]J. Tate. Nilpotent quotient groups.Topology,3(1964):109-111.
[131]J.G. Thompson, Normal ρ-complements for finite groups, J. Alge-bra.1(1964),43-46.
[132]J.G. Thompson. Nonsolvable finite groups adl of whose local sub-groups are solvable I. Bull. Amer. Math. Soc.,74(1968):104-109.
[133]王殿军,用极大子群阶之集刻画有限单群,西南师范大学学报(自然科学版),1(1993),18-21.
[134]J. Wang, X. Guo, On the Norm of Finite Groups, Algebra Collo-quium,4(2007) 605-612.
[135]L. Wang and Y. Wang. Finite groups with transitive semiper-mutability. Front. Math. China,3(2008):101-108.
[136]Y. Wang, H. Wei and Y. Li. A generalization of Kramer's theorem and its Applications. Bull. Austral. Math. Soc.,65(2002):46-475.
[137]Y. Wang. c-normality of groups and its properties. J. Algebra, 180(1996) :954-965.
[138]Y. Wang. Finite groups with some subgroups of Sylow subgroups c-supplemented. J. Algebra,224(2000):467-478.
[139]B.A.F. Wehrfritz, Infinite Linear Groups, Springer-Verlag, Berlin 1973.
[140]B.A.F. Wehrfritz, Conditions for linear groups to have unipotent derived subgroups, preprint 2010.
[141]H. Wei and Y. Wang. On c* -normality and its properties. J. Group Theory,10(2007):211-223.
[142]H. Wei, Y. Wang and Y. Li. On c-supplemented maximal and min-imal subgroups of Sylow subgroups of finite groups. Proc.Amer.Math.Soc.,2004,132(8):2197-2204.
[143]H. Wielandt. Uber das Product Yon paarweise vertauschbaren nilpotenten Gruppen. Math. Z.,1951,55:1-7.
[144]H. Wielandt. U ber der Normalisator der Subnormalen Untergrup-pen, Math. Z.,69(1958):463-465.
[145]徐明曜.有限群导引(上册)(第二版).北京：科学出版社,2001.
[146]Mingyao Xu, Another proof of the solvability of finite groups with at most two conjugacy classes of maximal subgroups, Chin. Ann. Math.,6B (1985),211-213.
[147]徐明曜.有限群导引(下册)(第二版).北京：科学出版社,2001.
[148]G. Zacher. I gruppi risolubli in cui i sottogruppi di composizione coincidono con i sottogrupi quasinormali. Atti Accad. Naz. Lincei Rend.cl.Sci.Fis.Mat.Natur.,1964,37(8):150-154.
[149]曾吉文,具有TI亏群块的Alperin猜想问题,数学年刊,19(1998)：519-524.
[150]曾吉文,张继平.Brauer 21问题.中国科学(A辑),31(2001)：785-790.
[151]G. Zhang, On two theorems of Thompson, Proc. Amer. Math. Soc., 98 (4)(1986),579-582.
[152]J. Zhang, A note on finite groups satbfying the permutizer condi-tion, Science Bulletin 31(1986),363-365.
[153]J. Zhang, Finite groups with many conjugate elements, J. Algebra, 170(1994):608-624.
[154]张继平,关于有限群的Syskin问题,中国科学(A),18(2)(1988)：124-128.
[155]Q. Zhang. Finite groups with only seminormal and abnormal sub-groups. J. Math. Study 29(1996):10-15.
[156]Q. Zhang. S-semipermutablity and abnormality in finite groups. Comm. Algebra,27(1999):4515-4524.
[157]Q. Zhang, X. Guo, H. Qu and M. Xu, Finite groups which have many normal subgroups, J. Korean Math. Soc.46(2009):1165-1178
[158]张志让,无限群基础,西南师范大学出版社,1998.
[159]张志让,对于某些素数共轭类长无平方因子的有限群,数学进展34(2005)：93-101.
[2]M.J. Alejandre, A. Ballester-Bolinches and M.C. Pedraza-Aguilera. Finite soluble groups with permutable subnormal subgroups. J. Algebra,240(2001):705-722.
[3]Z. Arad, M.B. Ward. New criteria for the solvability of finite groups. J.Algebra,77(1982):234-246.
[4]M. Asaad. On the supersolvability of finite groups. Annales Univ. Sci.Budapest,ⅫⅠ,18(1975):3-7.
[5]M. Asaad, p-Nilpotence and Supersolvable of Finite Groups, Com-munications in Algebra,34(2006):189-195.
[6]M. Asaad and A.A.Heliel.On S-quasinormal embedded subgroups of finite groups.J. Pure Appl. Algebra,165(2001):129-135.
[7]M. Assad, M. Ramadan and A. Shaalan. Infuence of S-quasinormality on maximal subgroups of Sylow subgroups of Fitting subgroups of a finite group. Arch.Math.,1991,56:521-527.
[8]R. Baer. Supersoluble immersion. Cana. J. Math.,11(1959):353-369.
[9]R. Baer, Der Kern eine charakteristische Untergruppe. Compositio Math.1(1934):254-283.
[10]R. Baer, Norm and hypernorm, Publ. Math. Debrecen,4(1956), 347-356.
[11]R. Baer, Classes of finite groups and their properties, Illinois J. Math,1 (1957),115-187.
[12]R. Baer, Zentrum und Kern von Gruppen mit Elementen un-endlicher Ordnung, Compositio Math.2 (1935) 247-249.
[13]A. Ballester-Boliches. Permutably embedded subgroups of finite sol-uble groups. Arch. Math.,65(1995):1-7.
[14]A. Ballester-Bolinches and M.C. Pedraza-Aquilera. Sufficient condi-tions for supersolvability of finite groups. J. Pure Appl.Algebra, 127(1998):223-228.
[15]A. Ballester-Bolinehes, Y. Wang and X. Guo. c-Supplemented sub-groups of finite groups, Glasgow Math. J.,42(2000):383-389.
[16]A. Ballester-Bolinches, R. Esteban-Romero, On finite T-groups. J. Aust. Math. Soc.,75(2003):181-191.
[17]S. Bauman and G. Ebter. A note on subnormal and abnormal chains. J. Algebra,36(1975):287-293.
[18]B. Baumeister. A characterization of the finite soluble groups. Arch. Math.,72(1999):167-176.
[19]J.C. Beidleman, B. Brewster and D. J.S. Robinson. Criteria for per-mutability to be transitive in finite groups. J. Algebra,222(1999): 400-412.
[20]J.C. Beidleman and H. Heineken. Finite soluble groups whose sub-normal subgroups permute with certain class of subgroups. J. Group Theorey,6(2003):139-158.
[21]J.C. Beidleman, H. Heineken, M. Newell. Center and norm, Bull. Austral. Math. Soc.69(2004) 457-464.
[22]V.A. Belonogov. Finite groups with three classes of maximal sub-groups, Math.Sb.,131(173)(1986),225-239.
[23]Y. Berkovich, Z. Janko. Structure of finite ρ-groups with given sub-groups. Ischia group theory 2004,13-93, Contemp. Math.,402.
[24]Ja.G., Berkovic. The existence of subgroups of a finite non-solvable group, Dokl. Akad. Nauk 156(1964):1255-1257.
[25]P. Bhattacharya, N.P. Mukherjee. On the intersection of a class of maximal subgroups of a finite group, J. Pure and Applied Algebra, 42(1986),117-124.
[26]Bianchi, M., Mauri, A.G.B., Herzog, M., Verardi, L., On finite solv-able groups in which normality is a transitive relation. J. Group Theory 3(2000):147-156.
[27]Z. Bozikov and Z. Janko, A complete classification of finite p-groups all of whose noncyclic subgroups are normal, Glasnik Matematicki. 44(2009),177-185.
[28]R. Brandl, Groups with few non-normal subgroups, Comm. Algebra, 23(1995),2091-2098.
[29]K. Brown, Finite groups with nilpotent maximal A-invariant sub-groups, Arch. Math. (Basel),22(1971),146-149.
[30]R.A. Bryce and John Cossey, The Wielandt subgroup of a finite soluble group. J. London Math. Soc.40(1989),244-256.
[31]J. Buckley. Finite groups whose minimal subgroups are normal. Math.Z.,116(1967),15-17.
[32]A.R. Camina, The Wielandt length of finite groups. J. Algebra 15(1970),142-148.
[33]R.W. Carter. Nilpotent self-normalizing subgroups of suluble groups. Math. Z.,75(1961):136-139.
[34]Guiyun Chen, On the structure of Frobenius groups and 2-Frobenius groups, J. Southwest China Normal University,20:5 (1995),485487. (Chinese)
[35]Guiyun Chen, On Thompson's conjecture, J. Algebra,185(1996), 184-193.
[36]Guiyun Chen, A new characterization of PS2(q), Southeast Asian Bull. of Math.22(1998),257-263.
[37]C. Chen. Generalization of Schur-Zassenhus Theorem. J. Math. 18(1998):290-294.
[38]陈重穆,内外∑-群与极小非∑-群,重庆：西南师范大学出版社,1988.
[39]陈重穆,Sriniasan定理的一个推广.西南师范大学学报(自然科学版),1(1987)：1-4.
[40]J.H. Conway, et al, Atlas of Finite Groups, Oxford: Clarendon Press,1985.
[41]Piroska Csorgo, Marcel Herzog, On Supersolvable Groups and the Nilpotator. Comm. Algebra,32(2004):609-620.
[42]W.E.Deskins, A note on the index complex of a maximal subgroup, Arch Math.(Basel),54(1990),236-240.
[43]R. Doerk and T. Hawkes. Finite soluble groups. Berlin-New York, Walter De Gruyter,1992.
[44]Xuefu Duan, Group Theory, Proceedings of an International Sym-posium held in Beijing, Aug.27-Sep.8,1984.
[45]L.M. Ezquerro. A contribution to the theory of finite supersolv-able groups. Rendieonti del Seminario Matematico della Universitdi Padova,1993,89:161-170.
[46]樊恽,李伟,能表示成三个真子群的并集的群,华中师范大学学报(自然科学版),1(2008)：1-3.
[47]樊恽,郭秀云,岑嘉评.关于子群的两种广义正规性的注记.数学年刊,27(2006)：169-176.
[48]A. Fattahi. Groups with only normal and abnormal subgroups. J. Algebra,28(1974):15-19.
[49]W. Feit, J.G. Thompson, Solvability of groups of odd order, Pacific J. Math.,13 (1963),775-1029.
[50]Flores, Ramon J., Foote, Richard M., Strongly closed subgroups of finite groups, Advances in Mathematics 222(2009):453-484.
[51]T. Foguel. Conjugate-permutable subgroups. J. Algebra,191(1997): 235-239.
[52]W. Gaschiitz. Praefrattini gruppen. Arch. Math.,13(1962):418-426.
[53]W. Gaschiitz. Gruppen,in dennen das Normalteilersein transitiv ist. J. Reine Angew. Math.,198(1957):87-92.
[54]J.D. Gillam. Cover-avoid subgroups in finite solvable groups. J.Algebra,29(1974):324-329.
[55]G. Glauberman, On Burnside's paqb theorem. Pacific J. Math. 56(1975):469-476.
[56]D. Goldschmidt,2-fusion in finite groups. Ann. of Math.99(1974): 70-117.
[57]D. Goldschmidt, Strongly closed 2-subgroups of finite groups. Ann. of Math.102(1975):475-489.
[58]D. Gorenstein. Finite Groups. New York:Harper Row,1968.
[59]D. Gorenstein. Finite Simple Groups. New York: Plenum Press,1982.
[60]W. Guo, K.P. Shum and A. Skiba. G-covering subgroup sys-tems for the classes of supersolvable and nilpotent groups. Israel J.Math.,2003,183:125-138.
[61]W. Guo. On F-supplemented subgroups of finite groups. Manuscripta Math.,2008,127:139-150.
[62]W. Guo, K.P. Shum and A.N. Skiba. X-permutable maximal sub-groups of Sylow subgroups of finite groups. Ukrainian Mathematical Journal,2006,5s(10):1471-1480.
[63]W. Guo, Shum K. P., and Skiba A. N., X-quasinormal subgroups, Siberian Math. J.48(2007):593-605.
[64]郭秀云,非幂零极大子群指数为素数幂的有限群,数学年刊(A辑),6(1994),721-725.
[65]X. Guo, P. Guo, K.P. Shum, On semi cover-avoiding subgroups of finite groups, J. Pure Appl. Algebra,209(2007),151-158.
[66]X. Guo, S. Li, P. Flavell, Finite groups whose abelian subgroups are TI-subgroups, J. Algebra,307(2007),565-569.
[67]郭秀云,岑嘉评,有限群的极小子群与p-幂零性,中国科学(A辑),9(2002),782-790.
[68]X. Guo and K.P. Shum. Cover-avoidance properties and the struc-ture of finite groups. J. Pure Appl.Algebra,2003.181:297-308.
[69]R.M. Guralnick, Subgroups of prime-power index in a simple group, J. Algebra,81 (1983),304-311.
[70]M. Hall. The theory of group. New York:Chelesea Pub.Corn.,1976.
[71]P. Hall. A characteristic property of soluble groups. J.London Math. Soc.,12(1937):198-200.
[72]M. Herzog, On finite simple groups of order divisible by three primes only, J. Algebra,10(1968),383-388.
[73]B. Huppert. Endliche gruppen I.Berlin/ Heidelberg/ New York:Springer-Verlag,1967.
[74]B. Huppert and N. Blackburn. Finite Groups Ⅲ.Berlin/ New York:Springer-Verlag,1982.
[75]I.M. Isaacs, Character theory of finite groups, New York-London, 1976.
[76]N. Ito. Note on (LM)-groups of finite orders.Kodai Math.Seminar report,3(1951):1-6.
[77]Z. Janko, Endliche Gruppen mit lauter nilpotenten zweitmaximalen untergrupprn, Math. Zeitschr.79(1962):422-424.
[78]P. Jin, Automorphisms of groups, Journal of Algebra,312(2007): 562-569.
[79]O.H. Kegel. Sylow gruppen and subnormalteiler endlicher grup-pen.Math.Z.,78(1962):205-221.
[80]O.H. Kegel, B.A.F. Wehrfritz, Locally Finite Groups, North-Holland, Amsterdam 1973.
[81]E.I., Khukhro, V.D., Mazurov, Finite groups with an automorphism of prime order whose centralizer has small rank. Journal of Algebra, 301 (2006):474-492.
[82]H. Kurzweil, B. Stellmacher, The Theory of Finite Groups:An Introduction, Springer-Verlag, New York,2004.
[83]R. Lauer. Dualization for saturation for locally defined formation. J.Algebra,52(1978):347-353.
[84]J.C. Lennox and S.E.Stonehewer, Subnormal subgroups of groups, Oxford,1987.
[85]S. Li, On minimal non-PE-group. J. Pure Appl. Algebra,132 (1998):149-158.
[86]S. Li, On minimal subgroups of finite groups, Comm.Algebra, 22(1994):1913-1918.
[87]S. Li. Q. Zhang. A note on p-supersoluble groups. Acta Mathematica Hungar.,116(2007):27-34.
[88]李世荣,非正规极大子群同阶类类数等于2的有限群,数学学报,3(1990),388-392.
[89]S. Li and X. He. On normally embedded subgroups of prime power order in finite groups. Comm. Algebra,2008,36(6):2333-2340.
[90]S. Li and Y. Zhao, Some finite nonsolvable groups characterized by their solvable subgroups, Acta Math. Sinica, New Series,4(1988): 5-13.
[91]X. Li, A characterization of the finite simple groups, J. Algebra, 245(2001),620-649.
[92]黎先华,有限单群的一个刻画与极大子群的指数集合,中国科学(A辑),1(2004),48-60.
[93]Y. Li and Y. Wang. On S-pquasinormally embedded subgroups of finite groups, J.Algebra,281(2004):109-123.
[94]Y. Li and Y. Wang. The influence of minimal subgroups on the structure of a finite group. Proc. Amer. Math.Soc.,2003,131(2):337-341.
[95]Y. Li, S. Qiao and Y. Wang. On weakly s-permutably embedded subgroups of finite groups. Comm. Algebra,2009,37:1086-1097.
[96]G.A. Miller and H.C. Moreno. Non-abelian groups in which every subgroup is abelian. Trans. Amer. Math. Soc.,4(1903):398-404.
[97]M. Miyamoto, Solvability of some groups, Hokkaido Math.J. 11(1982),106-110.
[98]H. Mousavi, On Finite groups with few non-normal subgroups, 27(1999),3143-3151.
[99]0. Ore. Contributions to the theory of groups of finite order. Duke Math. J.,5(1939):431-460.
[100]D.S. Passman, Nonnormal subgroups of p-groups, J.Algebra 15(1970),352-370.
[101]T.A. Peng, Finite groups with pronormal subgroups, Proc. Amer. Math. Soc.,20(1969):232-234.
[102]C.E. Praeger, Wujie Shi, A characterization of some alternating and symmetric groups, Comm. Algebra,5 (1994),1507-1530.
[103]G. Qian, F. Tang, Finite groups all of whose abelian subgroups are QTI-subgroups, J. Algebra,320 (2008),3605-3611.
[104]G. Qian, F. Tang, Finite groups all of whose subgroups are abelian or normal. Journal of Mathematical Research and Expo-sition,28(2008):273-278.
[105]G. Qian, Finite groups with n-maximal subgroup, Comm. Alge-bra, (in press)
[106]Ren Yongcai, Notes on S-quasinormal subgroups in finite groups, Proc. Amer. Math. Soc.,117(3)1993,631-636.
[107]D.J.S. Robinson. A note on finite groups in which normality is transitive. Proc. Amer. Math.Soc.,1968,19(4):933-937.
[108]D.J.S. Robinson. Groups which are minimal with respect to nor-mality being transitive. Pacific Journal of Math.31(1969):777-785.
[109]D.J.S. Robinson. A Course in the Theory of Groups (Second Edi-tion), Springer-Verlag, New York,1996.
[110]J.S. Rose. On finite insoluble groups with nilpotent maximal sub-groups, J. Algebra,48 (1977),182-196.
[111]N. Sastry. On minimal non-PN-groups. J. Algebra,65(1982):104-109.
[112]P. Schmid. Subgroups permutable with all Sylow subgroups. J. Algebra,1998,207:285-293.
[113]E. Schenkman. On the norm of a group, Illinois J. Math.4 (1960) 150-152.
[114]Shirong Li, Zhencai Shen, et. The influence of SS-quasinormality of some subgroups on the structure of finite groups. J. Algebra, 2008,319:4275-4287.
[115]Zhencai Shen, Wujie Shi, Finite groups with normally embedded subgroups, Joural of Group Theory 13(2010):257-265.
[116]Zhencai Shen, Wujie Shi, Qingliang Zhang. S-quasinormality of finite groups. Front. Math. (China), (5)(2010):329-339.
[117]Zhencai Shen, Shirong Li, Wujie Shi, Finite groups all of whose second maximal subgroups are PSC-groups. Journal of Algebra and Its Application.8(2009),229-242.
[118]Zhencai Shen, Shirong Li. On SS-quasinormal subgroups of finite groups. Communication In Algebra.36(2008):4436-4447.
[119]Shirong Li, Zhencai Shen. On the intersection of the normalizers of derived subgroups of all subgroups of a finite group, Journal of Algebra 323(2010),1349-1357.
[120]Zhencai Shen, Wujie Shi, Finite Groups with pronormal subgroups. Acta Math. Sinica, New Series (2011).
[121]Zhencai Shen, Wujie Shi, On SNS-permutable subgroups and the structure of finite groups. Publ. Math. Debrecen.78(2011):159-168.
[122]Zhencai Shen, Ni Du, Finite groups with H-subgroups, Algebra Collaq. (China),2012.
[123]施武杰,极大子群同阶类类数不大于2的有限群,数学年刊(A辑),5(1989),532-537.
[124]施武杰,关于单K4-群,科学通报,17(1991),1281-1283.
[125]Wujie Shi, A note on p-nilpotence of finite groups, J. Algebra, 241(2001):435-436.
[126]施武杰,毕建行,交错群的一个新刻画,科学通报,9(1989)：715-716.
[127]苏向盈.有限群的半正规子群.数学杂志,8(1988)：5-10.
[128]A.N. Skiba. On weakly s-permutable subgroups of finite groups. J.Algebra,315(2007):192-209.
[129]S. Srinivasan.Two sufficient conditions for supersolvability of finite groups. Israel J. Math.,35(1980):210-214.
[130]J. Tate. Nilpotent quotient groups.Topology,3(1964):109-111.
[131]J.G. Thompson, Normal ρ-complements for finite groups, J. Alge-bra.1(1964),43-46.
[132]J.G. Thompson. Nonsolvable finite groups adl of whose local sub-groups are solvable I. Bull. Amer. Math. Soc.,74(1968):104-109.
[133]王殿军,用极大子群阶之集刻画有限单群,西南师范大学学报(自然科学版),1(1993),18-21.
[134]J. Wang, X. Guo, On the Norm of Finite Groups, Algebra Collo-quium,4(2007) 605-612.
[135]L. Wang and Y. Wang. Finite groups with transitive semiper-mutability. Front. Math. China,3(2008):101-108.
[136]Y. Wang, H. Wei and Y. Li. A generalization of Kramer's theorem and its Applications. Bull. Austral. Math. Soc.,65(2002):46-475.
[137]Y. Wang. c-normality of groups and its properties. J. Algebra, 180(1996) :954-965.
[138]Y. Wang. Finite groups with some subgroups of Sylow subgroups c-supplemented. J. Algebra,224(2000):467-478.
[139]B.A.F. Wehrfritz, Infinite Linear Groups, Springer-Verlag, Berlin 1973.
[140]B.A.F. Wehrfritz, Conditions for linear groups to have unipotent derived subgroups, preprint 2010.
[141]H. Wei and Y. Wang. On c* -normality and its properties. J. Group Theory,10(2007):211-223.
[142]H. Wei, Y. Wang and Y. Li. On c-supplemented maximal and min-imal subgroups of Sylow subgroups of finite groups. Proc.Amer.Math.Soc.,2004,132(8):2197-2204.
[143]H. Wielandt. Uber das Product Yon paarweise vertauschbaren nilpotenten Gruppen. Math. Z.,1951,55:1-7.
[144]H. Wielandt. U ber der Normalisator der Subnormalen Untergrup-pen, Math. Z.,69(1958):463-465.
[145]徐明曜.有限群导引(上册)(第二版).北京：科学出版社,2001.
[146]Mingyao Xu, Another proof of the solvability of finite groups with at most two conjugacy classes of maximal subgroups, Chin. Ann. Math.,6B (1985),211-213.
[147]徐明曜.有限群导引(下册)(第二版).北京：科学出版社,2001.
[148]G. Zacher. I gruppi risolubli in cui i sottogruppi di composizione coincidono con i sottogrupi quasinormali. Atti Accad. Naz. Lincei Rend.cl.Sci.Fis.Mat.Natur.,1964,37(8):150-154.
[149]曾吉文,具有TI亏群块的Alperin猜想问题,数学年刊,19(1998)：519-524.
[150]曾吉文,张继平.Brauer 21问题.中国科学(A辑),31(2001)：785-790.
[151]G. Zhang, On two theorems of Thompson, Proc. Amer. Math. Soc., 98 (4)(1986),579-582.
[152]J. Zhang, A note on finite groups satbfying the permutizer condi-tion, Science Bulletin 31(1986),363-365.
[153]J. Zhang, Finite groups with many conjugate elements, J. Algebra, 170(1994):608-624.
[154]张继平,关于有限群的Syskin问题,中国科学(A),18(2)(1988)：124-128.
[155]Q. Zhang. Finite groups with only seminormal and abnormal sub-groups. J. Math. Study 29(1996):10-15.
[156]Q. Zhang. S-semipermutablity and abnormality in finite groups. Comm. Algebra,27(1999):4515-4524.
[157]Q. Zhang, X. Guo, H. Qu and M. Xu, Finite groups which have many normal subgroups, J. Korean Math. Soc.46(2009):1165-1178
[158]张志让,无限群基础,西南师范大学出版社,1998.
[159]张志让,对于某些素数共轭类长无平方因子的有限群,数学进展34(2005)：93-101.
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