某些子群对有限群结构的影响
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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.
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.
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.
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.
引文
1 Y. M. Wang. C-normality of Groups and Its Properties. Journal of Algebra,1996, 180: 954～965
2杨高才.关于几乎正规子群.山西大学学, 2007, 30(1): 17～20
3查明明,於遒.关于Sylow子群的极大子群的完全条件置换性.广西师院学报, 2005, 14(1): 4～6
4胡玉生,王丽丽.完全条件置换子群与有限群结构.山西师范大学学报, 2007,21(3): 1～4
5徐明耀.有限群导引(上册).北京:科学出版社, 1987: 1～181
6周凤玲.特殊子群对有限群结构的影响.哈尔滨师范大学硕士论文. 2009: 6～7
7陶司兴,王品超.幂零群的若干充分条件.商丘师范学院学报, 2006, 22(5):33～35
8唐曾林,钟祥贵,蒙忠传.幂零群的半正规, c-正规刻画.广西师范大学学报,2006, 23(4): 9～12
9 T. Foguel. Conjugate-Permutable Subgroups. Journal of Algebra, 1997, 191(1):235～239
10裴旭莲,钟祥贵.共轭置换子群与群的幂零性.广西师范大学学报, 2007,25(3): 28～31
11唐曾林,李世荣,曾凡辉.半正规, c-正规与有限群的超可解性.广西科学,2007, 14(1): 1～5
12刘仕田,刘自山. c-正规与有限群的p-幂零性.四川理工学院学报, 2007,20(5): 67～69
13 L. J. Zhu, W. B. Guo, K. P. Shum. Weakly C-normal Subgroups of FiniteGroups and Their Properties. Communications in Algebra, 2002, 33(11): 5505～5512
14焦学磊,史海军,郑志静.有限群可解的几个条件.四川理工学院学报, 2007,20(6): 8～10
15 S. R. Li. On Minimal Subgroups of Finite GroupsIII. Comm. Algebra, 1998,26(8): 453～461
16 M. Y. Xu, Q. H. Zhang. On Conjugate-Permutable Subgroups of a FiniteGroups. Algebra Colloquium, 2005, 12(4): 669～676
17 Q. H. Zhang, L. F. Wang. Finite Nonabelian Simple Groups which Contain aNontrivial Semipermutable Subgroup. Algebra Colloquium, 2005, 12(2): 301～307
18 R. Baer. Classes on Finite Groups and Their Properties. Illinois J Math,1957, 1: 115～187
19 S. R. Li, Z. C. Meng. Groups with Conjugate Permutable Conditions. Mathemat-ical Proceedings of the Royal Irish Academy, 2007, 107A: 115-121
20 W. B. Guo, K. P. Shum, A. N. Skiba. Criterions of Super Solubility for Productsof Supersoluble Groups. Siberian Math, 2004, 45(1): 115～128
21 Y. M. Wang. Finite Groups with Some Subgroups of Sylow Subgroups C-Supple-mented. J Algebra, 2004, 224: 467～478
22 D. Li, X. Y. Guo. On Complemented Subgroups of Finite Groups. Chin Annof Math, 2001, 22B(2): 249～254
23 H. G. Wei. On C-normal Maximal and Minimal Subgroups of Sylow Subgoupsof Finite Groups. Communications in Algebra, 2001, 29(5): 2193～2200
24М.И.Каргаполов,Ю.И.Мерзляков.群论基础.邓应生.高等教育出版社, 1994: 1～6
25 X. Y. Guo, K. P. Shum. Coveravo Idance Properties and the Structure ofFinite Groups. J Pure and Applied Algebra, 2003, 181: 297～308
26 D. Y. Li, X. Y. Guo. The In?uence of Normality of Subgroups on the Structureof Finite Groups. J Pure and Applied Algebra, 2000, 150: 53～60
27 K. Doerk. Minimal Nicht Uberauilosbare. Endliche Gruppen Math, 91(1966):198～205
28 O. Ore. Contributions to The Theory of Groups. Duke Math J, 1939, 5:431～460
29 S. G. Li. On The Centralizers of Minimal Subgroups of Finite Groups. Bollet-tino U M I, 1994, B-A(7): 439～447
30 Y. M. Wang. C-normality and Solvability of Groups. Journal of Pure and Ap-plied Algebra, 1996, 110: 315～320
31薛瑞,王品超.有限群超可解的若干充分条件.扬州大学学报, 2005, 8(4):9～12
32刘熠,曾意,王坤仁. s-拟正规子群对有限群结构的影响.四川师范大学学报,2009, 32(1): 39～42
33钱国华,朱平天.超可解群的一些充分条件.南京师大学报, 1998, 21(1):15～17
34陈重穆.内外-∑群与极小非∑群.重庆:西南师范大学出版社, 1988:1～27
35张远达.有限群构造.北京:科学出版社, 1982: 32～33
36何鸣,张雪梅,繆龙.子群的π-可补性对群结构的影响.扬州大学学报, 2005,8(1): 1～3
37高建玲,王丽芳. p-幂零群的若干充分条件.数学研究, 2008, 41(2): 163～167
38陈顺民,张良才.关于超可解群的若干结果.南京师大学报, 2005, 28(2):33～37
39张远达.幂零与可解之间.武汉:武汉大学出版社, 1988: 1～366
40王丽芳. s-半置换子群对群的幂零性的影响.山西师范大学学报, 2006,20(4): 6～9
41赵勇.子群s-正规性对群结构的影响.四川师范大学学报, 2007, 30(3):280～283
42高建玲.共轭置换子群对群p-幂零性的影响.山西大同大学学报, 2008,24(6): 12～14
43胡玉生,郭秀云.条件置换子群对有限群结构的影响.上海大学学报, 2007,13(1): 28～32
2杨高才.关于几乎正规子群.山西大学学, 2007, 30(1): 17～20
3查明明,於遒.关于Sylow子群的极大子群的完全条件置换性.广西师院学报, 2005, 14(1): 4～6
4胡玉生,王丽丽.完全条件置换子群与有限群结构.山西师范大学学报, 2007,21(3): 1～4
5徐明耀.有限群导引(上册).北京:科学出版社, 1987: 1～181
6周凤玲.特殊子群对有限群结构的影响.哈尔滨师范大学硕士论文. 2009: 6～7
7陶司兴,王品超.幂零群的若干充分条件.商丘师范学院学报, 2006, 22(5):33～35
8唐曾林,钟祥贵,蒙忠传.幂零群的半正规, c-正规刻画.广西师范大学学报,2006, 23(4): 9～12
9 T. Foguel. Conjugate-Permutable Subgroups. Journal of Algebra, 1997, 191(1):235～239
10裴旭莲,钟祥贵.共轭置换子群与群的幂零性.广西师范大学学报, 2007,25(3): 28～31
11唐曾林,李世荣,曾凡辉.半正规, c-正规与有限群的超可解性.广西科学,2007, 14(1): 1～5
12刘仕田,刘自山. c-正规与有限群的p-幂零性.四川理工学院学报, 2007,20(5): 67～69
13 L. J. Zhu, W. B. Guo, K. P. Shum. Weakly C-normal Subgroups of FiniteGroups and Their Properties. Communications in Algebra, 2002, 33(11): 5505～5512
14焦学磊,史海军,郑志静.有限群可解的几个条件.四川理工学院学报, 2007,20(6): 8～10
15 S. R. Li. On Minimal Subgroups of Finite GroupsIII. Comm. Algebra, 1998,26(8): 453～461
16 M. Y. Xu, Q. H. Zhang. On Conjugate-Permutable Subgroups of a FiniteGroups. Algebra Colloquium, 2005, 12(4): 669～676
17 Q. H. Zhang, L. F. Wang. Finite Nonabelian Simple Groups which Contain aNontrivial Semipermutable Subgroup. Algebra Colloquium, 2005, 12(2): 301～307
18 R. Baer. Classes on Finite Groups and Their Properties. Illinois J Math,1957, 1: 115～187
19 S. R. Li, Z. C. Meng. Groups with Conjugate Permutable Conditions. Mathemat-ical Proceedings of the Royal Irish Academy, 2007, 107A: 115-121
20 W. B. Guo, K. P. Shum, A. N. Skiba. Criterions of Super Solubility for Productsof Supersoluble Groups. Siberian Math, 2004, 45(1): 115～128
21 Y. M. Wang. Finite Groups with Some Subgroups of Sylow Subgroups C-Supple-mented. J Algebra, 2004, 224: 467～478
22 D. Li, X. Y. Guo. On Complemented Subgroups of Finite Groups. Chin Annof Math, 2001, 22B(2): 249～254
23 H. G. Wei. On C-normal Maximal and Minimal Subgroups of Sylow Subgoupsof Finite Groups. Communications in Algebra, 2001, 29(5): 2193～2200
24М.И.Каргаполов,Ю.И.Мерзляков.群论基础.邓应生.高等教育出版社, 1994: 1～6
25 X. Y. Guo, K. P. Shum. Coveravo Idance Properties and the Structure ofFinite Groups. J Pure and Applied Algebra, 2003, 181: 297～308
26 D. Y. Li, X. Y. Guo. The In?uence of Normality of Subgroups on the Structureof Finite Groups. J Pure and Applied Algebra, 2000, 150: 53～60
27 K. Doerk. Minimal Nicht Uberauilosbare. Endliche Gruppen Math, 91(1966):198～205
28 O. Ore. Contributions to The Theory of Groups. Duke Math J, 1939, 5:431～460
29 S. G. Li. On The Centralizers of Minimal Subgroups of Finite Groups. Bollet-tino U M I, 1994, B-A(7): 439～447
30 Y. M. Wang. C-normality and Solvability of Groups. Journal of Pure and Ap-plied Algebra, 1996, 110: 315～320
31薛瑞,王品超.有限群超可解的若干充分条件.扬州大学学报, 2005, 8(4):9～12
32刘熠,曾意,王坤仁. s-拟正规子群对有限群结构的影响.四川师范大学学报,2009, 32(1): 39～42
33钱国华,朱平天.超可解群的一些充分条件.南京师大学报, 1998, 21(1):15～17
34陈重穆.内外-∑群与极小非∑群.重庆:西南师范大学出版社, 1988:1～27
35张远达.有限群构造.北京:科学出版社, 1982: 32～33
36何鸣,张雪梅,繆龙.子群的π-可补性对群结构的影响.扬州大学学报, 2005,8(1): 1～3
37高建玲,王丽芳. p-幂零群的若干充分条件.数学研究, 2008, 41(2): 163～167
38陈顺民,张良才.关于超可解群的若干结果.南京师大学报, 2005, 28(2):33～37
39张远达.幂零与可解之间.武汉:武汉大学出版社, 1988: 1～366
40王丽芳. s-半置换子群对群的幂零性的影响.山西师范大学学报, 2006,20(4): 6～9
41赵勇.子群s-正规性对群结构的影响.四川师范大学学报, 2007, 30(3):280～283
42高建玲.共轭置换子群对群p-幂零性的影响.山西大同大学学报, 2008,24(6): 12～14
43胡玉生,郭秀云.条件置换子群对有限群结构的影响.上海大学学报, 2007,13(1): 28～32