有限群的Schmidt覆盖及其HS子群
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摘要
有限群的子群在研究有限群的性质及其结构中起着非常大的作用。我们常见的方法有利用有限群的极大子群、2-极大子群、以至n-极大子群来揭示有限群结构和性质。所以具有某些性质的子群与有限群的关系成为人们普遍关心的问题。本文将以有限群的一族非幂零子群、非超可解子群为出发点,探求群的结构及素因子个数的范围,得出了一些新的结果,完善并推进了这一领域的研究。
第三章我们定义了有限群的Schmidt覆盖,即群G的一族满足某些条件的非幂零真子群组成的集合。我们的结果丰富了关于极大子群、2-极大子群的一些重要定理,而且推广了O.J.Schmidt关于内幂零群的定理。第四章我们在Schmidt覆盖的基础上,定义了有限群的广义Schmidt覆盖的概念。我们不仅推广了O.J.Schmidt关于内幂零群的定理,而且还推广了J.S.Rose研究的非正规的极大子群均幂零的有限群。第五章我们引入了有限群的非超可解覆盖的概念,即群G的一族满足某些条件的非超可解真子群组成的集合,得到了关于群的可解性的一个充分条件,推广了B.Huppert关于内超可解群的定理。第六章我们引入了(?)S-子群的概念,推广了魏先彪博士与郭秀云教授提出的(?)C-子群的概念,得到了有限群p-幂零及超可解性性的一些充分必要条件。
It is well known that subgroups of finite groups play an important role ininvestigating the structure and properties of finite groups. For instance, ourusual methods to investigate the structure and properties of a group are usingmaximal subgroups,2-maximal subgroups even n-maximal subgroups. Hencestudying finite groups by subgroups with certain properties is widely concerned.In this thesis, we investigate the structure of finite groups under the assumptionof partial subgroups. Not only many known results are generalized, but also thestudy of maximal subgroups is perfected.
In Chapter3we investigate the structure of finite groups by a class of non-nilpotent proper subgroups under some assumption, which is called a Schmidtcovering. Our results enrich some important theorems on maximal subgroups and2-maximal subgroups, and also generalize the known results of O. J. Schmidt onminimal non-nilpotent groups. In Chapter4, based on the conception of Schmidtcovering of a finite group, we define generalized Schmidt covering of finite groups.It is a proper generalization of the class of groups whose maximal subgroups areeither nilpotent or normal, which was investigated by J. S. Rose. In Chapter5we introduce the concept of non-supersolvable covering, which is composed bycertain proper non-supersolvable subgroups with certain conditions. Our resultgeneralize the known results of B. Huppert on minimal non-supersolvable groups.In Chapter6we give the definition of H S-subgroups of finite groups. It is ageneralization of H C-subgroups, which is proposed by Doctor X. B. Wei andProfessor X. Y. Guo. Also, we get some necessary and sufcient conditions onp-nilpotence and supersolvability of finite groups.
第三章我们定义了有限群的Schmidt覆盖,即群G的一族满足某些条件的非幂零真子群组成的集合。我们的结果丰富了关于极大子群、2-极大子群的一些重要定理,而且推广了O.J.Schmidt关于内幂零群的定理。第四章我们在Schmidt覆盖的基础上,定义了有限群的广义Schmidt覆盖的概念。我们不仅推广了O.J.Schmidt关于内幂零群的定理,而且还推广了J.S.Rose研究的非正规的极大子群均幂零的有限群。第五章我们引入了有限群的非超可解覆盖的概念,即群G的一族满足某些条件的非超可解真子群组成的集合,得到了关于群的可解性的一个充分条件,推广了B.Huppert关于内超可解群的定理。第六章我们引入了(?)S-子群的概念,推广了魏先彪博士与郭秀云教授提出的(?)C-子群的概念,得到了有限群p-幂零及超可解性性的一些充分必要条件。
It is well known that subgroups of finite groups play an important role ininvestigating the structure and properties of finite groups. For instance, ourusual methods to investigate the structure and properties of a group are usingmaximal subgroups,2-maximal subgroups even n-maximal subgroups. Hencestudying finite groups by subgroups with certain properties is widely concerned.In this thesis, we investigate the structure of finite groups under the assumptionof partial subgroups. Not only many known results are generalized, but also thestudy of maximal subgroups is perfected.
In Chapter3we investigate the structure of finite groups by a class of non-nilpotent proper subgroups under some assumption, which is called a Schmidtcovering. Our results enrich some important theorems on maximal subgroups and2-maximal subgroups, and also generalize the known results of O. J. Schmidt onminimal non-nilpotent groups. In Chapter4, based on the conception of Schmidtcovering of a finite group, we define generalized Schmidt covering of finite groups.It is a proper generalization of the class of groups whose maximal subgroups areeither nilpotent or normal, which was investigated by J. S. Rose. In Chapter5we introduce the concept of non-supersolvable covering, which is composed bycertain proper non-supersolvable subgroups with certain conditions. Our resultgeneralize the known results of B. Huppert on minimal non-supersolvable groups.In Chapter6we give the definition of H S-subgroups of finite groups. It is ageneralization of H C-subgroups, which is proposed by Doctor X. B. Wei andProfessor X. Y. Guo. Also, we get some necessary and sufcient conditions onp-nilpotence and supersolvability of finite groups.
引文
[1] M. Asaad, On p-nilpotence and supersolvability of finite groups, Comm. Algebra,34(2006),189-195.
[2] M. Aschbacher, On the maximal subgroups of the finite classical groups, Inven-tiones,76(1984),496-514.
[3] R. Baer, Situation der untergruppen und struktur der gruppe, S. B. Heidelberg.Akad. Wiss.,2(1933),12-17.
[4] A. Ballester-Bolinches and R. Esteban-Romero, On minimal non-supersolvablegroupsS. Rev. Mat. Iberoamericana,23(2007),127-142.
[5] A. Ballester-Bolinches and R. Esteban-Romero and M. Asaad, Products of finitegroups, de Gruyter Expositions in Mathematics, Walter de Gruyter, Berlin,2010.
[6] A. Ballester-Bolinches and J. Cossey and R. Esteban-Romero, Groups with allmaximal subgroups supersoluble or normal, preprint.
[7] A. Ballester-Bolinches and L.M. Ezquerro, Classes of Finite Groups, Springer,Dordrecht, Netherlands,2006.
[8] A. Ballester-Bolinches, Y. M Wang and X. Y Guo, c-supplemented subgroups offinite groups, Comm. Algebra,42(2000),383-389.
[9] M. Bianchi and A. Gillio Berta Mauri and M. Herzog and L. Verardi, On finite solv-able groups in which normality is a transitive relation, J. Group Theory,3(2000),147-156.
[10] D. Buchthal, On factorized groups., Trans. Amer. Math. Soc.,183(1973),425-432.
[11] Y. Z. Cao and M. Mignotte, On simple K4-groups, Journal of Algebra,241(2001),658-668.
[12] R. W. Carter, Nilpotent self-normalizing subgroups of soluble groups, Math. Z.,75(1961),136-139.
[13] R. W. Carter, Simple Groups of Lie Type, Pure Appl. Math., Wiley, London1972.
[14] J. H. Conway and R. T. Curtis and S. P. Norton and R. A. Parker and R. A.Wilson, Atlas of Finite Groups, Oxford Univ. Press, London,1985.
[15] P. Cso¨rg¨o and M. Herzog, On supersolvable groups and the nilpotator, Comm.Algebra,32(2004),609-620.
[16] K. Doerk, Minimal nicht u¨beraufl¨osbare, endliche Gruppen, Math. Z.,91(1966),198-205.
[17] K. Doerk and T. Hawkes, Finite Soluble Groups, Number4in De Gruyter Expo-sitions in Mathematics. Walter de Gruyter, Berlin, New York,1992.
[18] W. Feit and J. G. Thompson, Solvability of groups of odd order. Pacific J. Math.,13(1963),775-1029.
[19] W. Gaschu¨tz, Uber die Φ-Untergruppen endlicher Gruppen, Math. Z.,58(1953),160-170.
[20] M. Giudici, Maximal subgroups of almost simple groups with socle PSL(2, q),preprint.
[21] D. Gorenstein, Finite Groups. Harper and Row Publishers, New York, Evanston,London,1968.
[22] D. Gorenstein, Finite Groups, Chelsea Pub. Co., New York,1980.
[23] X. Y. Guo and K. P. Shum, On c-normal maximal and minimal subgroups of Sylowp-subgroups of finite groups, Chinese J. Contemporary Math.,80(2003),561-569.
[24] X. Y. Guo and K. P. Shum, Finite p-nilpotent group with some subgroups c-supplemented, J. Aust. Math. Soc.,78(2005),429-439.
[25] B. Hartley, Sylow p-subgroups and local p-solvability. Journal of Algebra,23(1972),347-369.
[26] M. Herzog, On finite simple groups of order divisible by three primes only, Journalof Algebra,10(1968),383-388.
[27] B. Huppert, Normalteiler und maximale Untergruppen endlicher Gruppen, Math-ematische Zeitschift,60(1954),409-434.
[28] B. Huppert, Subnormale untergruppen und p-Sylowgruppen, Acta, Sci. Math.Szged,22(1961),46-61.
[29] B. Huppert, Endliche Gruppen I. Springer-Verlag, Berlin, Heidelberg, New York,1967.
[30] B. Huppert and N. Blackburn, Finite Groups II. Springer-Verlag, Berlin, NewYork,1982.
[31] B. Huppert and N. Blackburn, Finite Groups III. Springer-Verlag, Berlin, NewYork,1982.
[32] B. Huppert and W. Lempken, Simple groups of order divisible by at most fourprimes, Izv. Gomel. Gos. Univ. Im. F. Skoriny,3(2000),64-75.
[33] N. Ito, Note on LM-groups of finite orders, Kodai Math. Sem. Rep.,(1951),1-6.
[34] K. Iwasawa,U¨ ber die Struktur der endlichen Gruppen, deron echte Untergruppensamtlich nilpotent sind, Proc. Phys.-Math. Soc. Japan,(1941),1-4.
[35] A. Jafarzadeh and A. Iranmanesh, On simple Kn-groups for n=5,6, LondonMathematical Society Lecture Note Series,2005.
[36] L. S. Kazarin, Groups that can be represented as a product of two solvable sub-groups.(Russian) Comm. Algebra,14(6)(1986),1001-1066.
[37] O. H. Kegel, Sylow-gruppen und subnormalteiler endlicher gruppen, Math.Z.,(1962),202-221.
[38] H. Kurzweil and B. Stellmacher, The Theory of finite groups, an introduction.Springer-Verlag, Berlin, New York, Heidelberg,2004.
[39] P. Kleidman, The lower-dimensional finite classical groups and their subgroups,to appear in Longman Research Notes.
[40] P. Kleidman and M. W. Liebeck, The subgroup structure of the finite classicalgroups. Cambridge Univ. Press, London Math. Soc. Lecture Notes Series, Cam-bridge, UK,1990.
[41] C. H. Li and H. Zhang, The finite primitive groups with soluble stabilizers, andthe edge-primitive s-arc transitive graphs, Proc. London Math. Soc., to appear.
[42] Q. L. Li and X. Y. Guo, On p-nilpotence and solubility of groups, Arch. Math.,96(2011),1-7.
[43] S. Li, Z. Shen, J. Liu and X. Liu, The influence of SS-quasinormality of somesubgroups on the structure of finite groups, Journal of Algebra,219(2008),4275-4287.
[44] S. Li and W. Shi, A note on the solvability of groups, Journal of Algebra,304(2006),278-285.
[45] T. Z. Li, Finite groups in which sylow normalizers admit supplements, Acta Math-ematica Sinica, English Series Mar.,23(3)(2007),417-422.
[46] M. W. Liebeck and C. E. Praeger and J. Saxl, The maximal factorizations of the fi-nite simple groups and their automorphism groups. Amer. Math. Soc. Providence,USA,1990.
[47] Y. Li and G. Wang and H. Wei, The influence of π-quasinormality of some sub-groups of a fintie group, Arch. Math.,81(2003),245-252.
[48] I. Martin Isaacs, Finite Group Theory, American Mathematical Society Provi-dence, Rhode Island,(2008).
[49] S. J. Nickerson, Maximal subgroups of the sporadic almost simple groups, Univer-sity of Birmingham,2003.
[50] O. Ore, Contributions to the theory of groups, Duke Math. J.,5(1939),431-460.
[51] D. J. S. Robinson, A Course in the Theory of Groups. Springer-Verlag, New York,1982.
[52] J. S. Rose, The influence on a finite group of its proper abnormal structure, J.London Math. Soc.,40(1965),348–361
[53] J. Rose, On finite insoluble groups with nilpotent maximal subgroups. Journal ofAlgebra,48(1977),182-196.
[54] M. Suzuki, On a class of doubly transitive groups, The Annals of Mathematics,75(1962),105-145.
[55] O. J. Schmidt,U¨ber gruppen, deren s¨amtliche teiler spezielle gruppen sind, Mat.Sbornik,31(1924),366-372.
[56] W. J. Shi, On simple K4-groups, Chinese Sci. Bull.,36(17)(1991),1281-283.
[57] L. I. Shidov, On maximal subgroups of finite groups, Sib. Matem. Zh.,3(1971),682-683.
[58] V. V. Shlyk, On the intersection of maximal subgroups in finite groups, Matem-aticheskie Zametki,14(3)(1973),429-439.
[59] A. N. Skiba, On weakly s-permutable subgroups of finite groups. Journal of Alge-bra,315(2007),192-209.
[60] J. Tate, Nilpotent quotient groups. Topology,3(1964),109-111.
[61] M. C. Tamburini and E. P. Vdovin, Carter subgroups of finite groups, Journal ofAlgebra,255(1)(2002),148-163.
[62] E. P. Vdovin, On the conjugacy problem for Carter subgroups. Sib. Mat. Zh.,47(4)(2006),597-600.
[63] E. P. Vdovin, Carter subgroups of finite almost simple groups. Algebra and Logic,46(2)(2007),90-119.
[64] Y. M. Wang, c-normality of groups and its properties, Journal of Algebra,180(1996),954-965.
[65] Y. M. Wang, H. Q. Wei and Y. M. Li, A generalisation of Kramer’s theorem and itsapplications, Bulletin of the Australian Mathematical Society,65(2002),467-475.
[66] X. B. Wei and X. Y. Guo, On H C-subgroups and the influence on the structureof finite groups, to appear.
[67] H. Q. Wei and Y. M. Wang and Y. M. Li, On c-normal maximal and minimalsubgroups of Sylow subgroups of finite groups. II, Comm. Algebra,31(2003),4807-4816.
[68] M. Weinstein(Ed), Between nilpotent and solvable, Polygonal publishing House,Passaic, NJ,1982.
[69]徐明曜,有限群导引(上册).北京:科学出版社,1987.
[70]徐明曜,黄建华,李慧陵,李世荣,有限群导引(下册).北京:科学出版社,2001.
[2] M. Aschbacher, On the maximal subgroups of the finite classical groups, Inven-tiones,76(1984),496-514.
[3] R. Baer, Situation der untergruppen und struktur der gruppe, S. B. Heidelberg.Akad. Wiss.,2(1933),12-17.
[4] A. Ballester-Bolinches and R. Esteban-Romero, On minimal non-supersolvablegroupsS. Rev. Mat. Iberoamericana,23(2007),127-142.
[5] A. Ballester-Bolinches and R. Esteban-Romero and M. Asaad, Products of finitegroups, de Gruyter Expositions in Mathematics, Walter de Gruyter, Berlin,2010.
[6] A. Ballester-Bolinches and J. Cossey and R. Esteban-Romero, Groups with allmaximal subgroups supersoluble or normal, preprint.
[7] A. Ballester-Bolinches and L.M. Ezquerro, Classes of Finite Groups, Springer,Dordrecht, Netherlands,2006.
[8] A. Ballester-Bolinches, Y. M Wang and X. Y Guo, c-supplemented subgroups offinite groups, Comm. Algebra,42(2000),383-389.
[9] M. Bianchi and A. Gillio Berta Mauri and M. Herzog and L. Verardi, On finite solv-able groups in which normality is a transitive relation, J. Group Theory,3(2000),147-156.
[10] D. Buchthal, On factorized groups., Trans. Amer. Math. Soc.,183(1973),425-432.
[11] Y. Z. Cao and M. Mignotte, On simple K4-groups, Journal of Algebra,241(2001),658-668.
[12] R. W. Carter, Nilpotent self-normalizing subgroups of soluble groups, Math. Z.,75(1961),136-139.
[13] R. W. Carter, Simple Groups of Lie Type, Pure Appl. Math., Wiley, London1972.
[14] J. H. Conway and R. T. Curtis and S. P. Norton and R. A. Parker and R. A.Wilson, Atlas of Finite Groups, Oxford Univ. Press, London,1985.
[15] P. Cso¨rg¨o and M. Herzog, On supersolvable groups and the nilpotator, Comm.Algebra,32(2004),609-620.
[16] K. Doerk, Minimal nicht u¨beraufl¨osbare, endliche Gruppen, Math. Z.,91(1966),198-205.
[17] K. Doerk and T. Hawkes, Finite Soluble Groups, Number4in De Gruyter Expo-sitions in Mathematics. Walter de Gruyter, Berlin, New York,1992.
[18] W. Feit and J. G. Thompson, Solvability of groups of odd order. Pacific J. Math.,13(1963),775-1029.
[19] W. Gaschu¨tz, Uber die Φ-Untergruppen endlicher Gruppen, Math. Z.,58(1953),160-170.
[20] M. Giudici, Maximal subgroups of almost simple groups with socle PSL(2, q),preprint.
[21] D. Gorenstein, Finite Groups. Harper and Row Publishers, New York, Evanston,London,1968.
[22] D. Gorenstein, Finite Groups, Chelsea Pub. Co., New York,1980.
[23] X. Y. Guo and K. P. Shum, On c-normal maximal and minimal subgroups of Sylowp-subgroups of finite groups, Chinese J. Contemporary Math.,80(2003),561-569.
[24] X. Y. Guo and K. P. Shum, Finite p-nilpotent group with some subgroups c-supplemented, J. Aust. Math. Soc.,78(2005),429-439.
[25] B. Hartley, Sylow p-subgroups and local p-solvability. Journal of Algebra,23(1972),347-369.
[26] M. Herzog, On finite simple groups of order divisible by three primes only, Journalof Algebra,10(1968),383-388.
[27] B. Huppert, Normalteiler und maximale Untergruppen endlicher Gruppen, Math-ematische Zeitschift,60(1954),409-434.
[28] B. Huppert, Subnormale untergruppen und p-Sylowgruppen, Acta, Sci. Math.Szged,22(1961),46-61.
[29] B. Huppert, Endliche Gruppen I. Springer-Verlag, Berlin, Heidelberg, New York,1967.
[30] B. Huppert and N. Blackburn, Finite Groups II. Springer-Verlag, Berlin, NewYork,1982.
[31] B. Huppert and N. Blackburn, Finite Groups III. Springer-Verlag, Berlin, NewYork,1982.
[32] B. Huppert and W. Lempken, Simple groups of order divisible by at most fourprimes, Izv. Gomel. Gos. Univ. Im. F. Skoriny,3(2000),64-75.
[33] N. Ito, Note on LM-groups of finite orders, Kodai Math. Sem. Rep.,(1951),1-6.
[34] K. Iwasawa,U¨ ber die Struktur der endlichen Gruppen, deron echte Untergruppensamtlich nilpotent sind, Proc. Phys.-Math. Soc. Japan,(1941),1-4.
[35] A. Jafarzadeh and A. Iranmanesh, On simple Kn-groups for n=5,6, LondonMathematical Society Lecture Note Series,2005.
[36] L. S. Kazarin, Groups that can be represented as a product of two solvable sub-groups.(Russian) Comm. Algebra,14(6)(1986),1001-1066.
[37] O. H. Kegel, Sylow-gruppen und subnormalteiler endlicher gruppen, Math.Z.,(1962),202-221.
[38] H. Kurzweil and B. Stellmacher, The Theory of finite groups, an introduction.Springer-Verlag, Berlin, New York, Heidelberg,2004.
[39] P. Kleidman, The lower-dimensional finite classical groups and their subgroups,to appear in Longman Research Notes.
[40] P. Kleidman and M. W. Liebeck, The subgroup structure of the finite classicalgroups. Cambridge Univ. Press, London Math. Soc. Lecture Notes Series, Cam-bridge, UK,1990.
[41] C. H. Li and H. Zhang, The finite primitive groups with soluble stabilizers, andthe edge-primitive s-arc transitive graphs, Proc. London Math. Soc., to appear.
[42] Q. L. Li and X. Y. Guo, On p-nilpotence and solubility of groups, Arch. Math.,96(2011),1-7.
[43] S. Li, Z. Shen, J. Liu and X. Liu, The influence of SS-quasinormality of somesubgroups on the structure of finite groups, Journal of Algebra,219(2008),4275-4287.
[44] S. Li and W. Shi, A note on the solvability of groups, Journal of Algebra,304(2006),278-285.
[45] T. Z. Li, Finite groups in which sylow normalizers admit supplements, Acta Math-ematica Sinica, English Series Mar.,23(3)(2007),417-422.
[46] M. W. Liebeck and C. E. Praeger and J. Saxl, The maximal factorizations of the fi-nite simple groups and their automorphism groups. Amer. Math. Soc. Providence,USA,1990.
[47] Y. Li and G. Wang and H. Wei, The influence of π-quasinormality of some sub-groups of a fintie group, Arch. Math.,81(2003),245-252.
[48] I. Martin Isaacs, Finite Group Theory, American Mathematical Society Provi-dence, Rhode Island,(2008).
[49] S. J. Nickerson, Maximal subgroups of the sporadic almost simple groups, Univer-sity of Birmingham,2003.
[50] O. Ore, Contributions to the theory of groups, Duke Math. J.,5(1939),431-460.
[51] D. J. S. Robinson, A Course in the Theory of Groups. Springer-Verlag, New York,1982.
[52] J. S. Rose, The influence on a finite group of its proper abnormal structure, J.London Math. Soc.,40(1965),348–361
[53] J. Rose, On finite insoluble groups with nilpotent maximal subgroups. Journal ofAlgebra,48(1977),182-196.
[54] M. Suzuki, On a class of doubly transitive groups, The Annals of Mathematics,75(1962),105-145.
[55] O. J. Schmidt,U¨ber gruppen, deren s¨amtliche teiler spezielle gruppen sind, Mat.Sbornik,31(1924),366-372.
[56] W. J. Shi, On simple K4-groups, Chinese Sci. Bull.,36(17)(1991),1281-283.
[57] L. I. Shidov, On maximal subgroups of finite groups, Sib. Matem. Zh.,3(1971),682-683.
[58] V. V. Shlyk, On the intersection of maximal subgroups in finite groups, Matem-aticheskie Zametki,14(3)(1973),429-439.
[59] A. N. Skiba, On weakly s-permutable subgroups of finite groups. Journal of Alge-bra,315(2007),192-209.
[60] J. Tate, Nilpotent quotient groups. Topology,3(1964),109-111.
[61] M. C. Tamburini and E. P. Vdovin, Carter subgroups of finite groups, Journal ofAlgebra,255(1)(2002),148-163.
[62] E. P. Vdovin, On the conjugacy problem for Carter subgroups. Sib. Mat. Zh.,47(4)(2006),597-600.
[63] E. P. Vdovin, Carter subgroups of finite almost simple groups. Algebra and Logic,46(2)(2007),90-119.
[64] Y. M. Wang, c-normality of groups and its properties, Journal of Algebra,180(1996),954-965.
[65] Y. M. Wang, H. Q. Wei and Y. M. Li, A generalisation of Kramer’s theorem and itsapplications, Bulletin of the Australian Mathematical Society,65(2002),467-475.
[66] X. B. Wei and X. Y. Guo, On H C-subgroups and the influence on the structureof finite groups, to appear.
[67] H. Q. Wei and Y. M. Wang and Y. M. Li, On c-normal maximal and minimalsubgroups of Sylow subgroups of finite groups. II, Comm. Algebra,31(2003),4807-4816.
[68] M. Weinstein(Ed), Between nilpotent and solvable, Polygonal publishing House,Passaic, NJ,1982.
[69]徐明曜,有限群导引(上册).北京:科学出版社,1987.
[70]徐明曜,黄建华,李慧陵,李世荣,有限群导引(下册).北京:科学出版社,2001.
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