关于有限群可解性的研究
|
摘要
有限群论是群论的基础部分，可解群是群论中一类比较常见的群，也是一类极其重要的群。许多群论专家已经得到诸多关于有限群可解的充分条件，有许多结论是研究有限群结构时有用的工具。本文的出发点就是在这些结论的基础上结合Sylow子群、Hall子群、共轭置换子群、c-正规子群等对有限群的可解性进行研究，得到以下主要结论：
(1)若G的Sylow 2-子群为交换群，且对G的任意Sylow 2-子群Q(Q≠P)，P∩Q在P中极大，则G为可解群。
(2)设G是偶阶群，P∈Syl_2(G)，若P在G中c-正规，则G为可解群。
(3)设M是G的极大子群而且是幂零群，如果M的Sylow 2-子群在G中是c-正规的，则G为可解群。
(4)设G是有限群，H是G的偶阶幂零Hall子群，M是H的极大子群，若M的Sylow 2-子群在G中是c-正规，则G是可解群。
(5)设H是G的偶阶π-Hall子群，若H及H的每个Sylow子群均在G中共轭置换，则G可解。
(6)设P为有限群G的Sylow P-子群，若P在G中共轭置换且G/P的极大子群为1，则G为可解群。
(7)设H是G的偶阶π-Hall子群，且H的每个Sylow子群都是正规的，又H在G中共轭置换，则G可解。
(8)设H是G的π-Hall子群，且2∈π，H幂零且在G中共轭置换，则G可解。
Finite group theory is the basic part of group theory. The solvable group is a kindof ordinary and important group. Many group experts have got some sufficiency aboutthe solvability, many of which are useful tools in researching the structure of finitegroups. In this paper, the starting point is researching the solvability, in the base ofthe conclusions,Combining Sylow-subgroups, Hall-subgroups, conjugate-permutablesubgroups and c-normal groups. We get the following conclusions:
(1) If G is sylow2-subgroups are Abelian, and P∩Q is enormous in P forany Sylow 2-subgroups Q(Q≠P),then G is solvable group.
(2) Let G be a group of even order, P∈Syl_2(G), if P is C-normal inG, then G is solvable group.
(3) Let M be a maximal subgroup and nilpotent group of G, if the Sylow2-subgroup of M in G is c-normal, then G is solvable group.
(4) Let G be a finite group, H is nilpotent Hall-subgroups with even order ofG. M's Sylow 2-subgroup is c-normal in G, then G is solvable group.
(5) Let H be aπ-Hall subgroup with even order of G, if H and its everySylow-subgroup are all conjugate-permutable, then G is solvable.
(6) Let P be a Sylow P-subgroup of G, If P is conjugate-permutable inG and G/P's maximal subgroup is 1, then G is solvable.
(7) Let H be aπ-Hall subgroup with even order of group G, if H'everySylow-subgroup of H is conjugate-permutable in G, then G is solvable.
(8) Let H beaπ-Hall subgroup of G, and 2∈π, if H is nilpotent andconjugate-permutable in G, then G is solvable.
(1)若G的Sylow 2-子群为交换群，且对G的任意Sylow 2-子群Q(Q≠P)，P∩Q在P中极大，则G为可解群。
(2)设G是偶阶群，P∈Syl_2(G)，若P在G中c-正规，则G为可解群。
(3)设M是G的极大子群而且是幂零群，如果M的Sylow 2-子群在G中是c-正规的，则G为可解群。
(4)设G是有限群，H是G的偶阶幂零Hall子群，M是H的极大子群，若M的Sylow 2-子群在G中是c-正规，则G是可解群。
(5)设H是G的偶阶π-Hall子群，若H及H的每个Sylow子群均在G中共轭置换，则G可解。
(6)设P为有限群G的Sylow P-子群，若P在G中共轭置换且G/P的极大子群为1，则G为可解群。
(7)设H是G的偶阶π-Hall子群，且H的每个Sylow子群都是正规的，又H在G中共轭置换，则G可解。
(8)设H是G的π-Hall子群，且2∈π，H幂零且在G中共轭置换，则G可解。
Finite group theory is the basic part of group theory. The solvable group is a kindof ordinary and important group. Many group experts have got some sufficiency aboutthe solvability, many of which are useful tools in researching the structure of finitegroups. In this paper, the starting point is researching the solvability, in the base ofthe conclusions,Combining Sylow-subgroups, Hall-subgroups, conjugate-permutablesubgroups and c-normal groups. We get the following conclusions:
(1) If G is sylow2-subgroups are Abelian, and P∩Q is enormous in P forany Sylow 2-subgroups Q(Q≠P),then G is solvable group.
(2) Let G be a group of even order, P∈Syl_2(G), if P is C-normal inG, then G is solvable group.
(3) Let M be a maximal subgroup and nilpotent group of G, if the Sylow2-subgroup of M in G is c-normal, then G is solvable group.
(4) Let G be a finite group, H is nilpotent Hall-subgroups with even order ofG. M's Sylow 2-subgroup is c-normal in G, then G is solvable group.
(5) Let H be aπ-Hall subgroup with even order of G, if H and its everySylow-subgroup are all conjugate-permutable, then G is solvable.
(6) Let P be a Sylow P-subgroup of G, If P is conjugate-permutable inG and G/P's maximal subgroup is 1, then G is solvable.
(7) Let H be aπ-Hall subgroup with even order of group G, if H'everySylow-subgroup of H is conjugate-permutable in G, then G is solvable.
(8) Let H beaπ-Hall subgroup of G, and 2∈π, if H is nilpotent andconjugate-permutable in G, then G is solvable.
引文
[1] 徐明曜．有限群导引(上、下)[M]，科学出版社．
[2] 陈重穆．有限群论基础[M]，重庆出版社．
[3] 王萼芳．有限群论基础[M]，清华大学出版社．
[4] 张远达．有限群构造(上、下)[M]，科学出版社．
[5]．熊全淹．近世代数[M]，武汉大学出版社．
[6] T. Foguel. Conjugate-permutable subgroups[J]. Journal of lgebral1997.
[7] WANG Y. c2normality of groups and its properties[J] Journal of algebra, 1996, 180:954-965.
[8] 王品超，温风桐．有限群的C一正规子群[J]．曲阜师范大学学报1997．
[9]. Hall M. The theory of groups, The Macmillan Co, New York. 1959.
[10] 张勤海，赵俊英．超可解群的若干充分条件[[J]．数学杂志，2005，25(4)：399-404．
[11] 郭鹏飞．共轭置换子群对有限群可解性的影响[[J]．山西农业大学学报，2004，24(2)：187-188．
[12] J. Tits, Groupes semi-simples isotropes Colloque sur la theorie des groupes algebriques[J], Bruxelles (1962).
[13] 王坤仁．子群皆半正规的有限群[J]，四川师范大学学报．1996．
[14] 李祥明．C一正规子群与有限群的结构[[J]．工程数学学报，2001(4)：88-92．
[15] P. C. Wang. Some sufficient conditions of a nilpotent group[J]. Journal of Algebra, 1992, 148: 289-295.
[16] J S. Rose. A course on groups theory [M]groupes lgebriques[J], Bruxelles (1962).
[17] H. Ward, On Ree's series of simple groups[J], Trans. Amer. Math. Soc. 121(1966).
[19] W. Wong, Generators and relations for classical groups[J], J. Algerbra 32 (1974).
[20] 张勤海，王俊新．子群为拟正规或自正规的有限群[J]，数学学报 1993．
[21] 张勤海．只具有半正规和反正规子群的有限群[J]，数学研究 1996．
[22] 李长稳，於遒，骆公志．可解群的一些充分条件[J]，淮阴师范学院学报 2004．
[23] 钟祥贵．有限群的某些sylow子群与可解性[J]，邵阳学院学报 2004．
[24] 李晓华．2-极大子群是次正规子群的有限群[J]，四川师范大学学报(自然科学版)2000．
[25] Zvonimir Janko, Elements of order at most 4 in finite 2-groups, Journal of Group Theory.
[26] Peter Schmid, On the Automorphism Group of Extraspecial 2-Groups, Journal of Algebra234, 492_506, 2000.
[27] Zvonimir Janko, A classification of finite 2-groups with exactly three involutions, Journal of Algebra 291 (2005) 505-533.
[28] Zvonirnir Janko, Finite 2-groups with a self-centralizing elementary abelian subgroup of order 8, Journal of Algebra 269 (2003) 189-214.
[29] M. F. Newman and Ming-Yao Xu, A classification of metacyclic p-groups, Research Report No, 32/1990.
[30] Yakov Berkovich, On Subgroups of Finite p-Groups, Journal of Algebra 224, 198]240_2000.
[31] Legovini P. Finite groups whose subgroups are either subnormal or pronormal Ⅱ[J], Rend Sem Math. Uni Padova, 1981.
[2] 陈重穆．有限群论基础[M]，重庆出版社．
[3] 王萼芳．有限群论基础[M]，清华大学出版社．
[4] 张远达．有限群构造(上、下)[M]，科学出版社．
[5]．熊全淹．近世代数[M]，武汉大学出版社．
[6] T. Foguel. Conjugate-permutable subgroups[J]. Journal of lgebral1997.
[7] WANG Y. c2normality of groups and its properties[J] Journal of algebra, 1996, 180:954-965.
[8] 王品超，温风桐．有限群的C一正规子群[J]．曲阜师范大学学报1997．
[9]. Hall M. The theory of groups, The Macmillan Co, New York. 1959.
[10] 张勤海，赵俊英．超可解群的若干充分条件[[J]．数学杂志，2005，25(4)：399-404．
[11] 郭鹏飞．共轭置换子群对有限群可解性的影响[[J]．山西农业大学学报，2004，24(2)：187-188．
[12] J. Tits, Groupes semi-simples isotropes Colloque sur la theorie des groupes algebriques[J], Bruxelles (1962).
[13] 王坤仁．子群皆半正规的有限群[J]，四川师范大学学报．1996．
[14] 李祥明．C一正规子群与有限群的结构[[J]．工程数学学报，2001(4)：88-92．
[15] P. C. Wang. Some sufficient conditions of a nilpotent group[J]. Journal of Algebra, 1992, 148: 289-295.
[16] J S. Rose. A course on groups theory [M]groupes lgebriques[J], Bruxelles (1962).
[17] H. Ward, On Ree's series of simple groups[J], Trans. Amer. Math. Soc. 121(1966).
[19] W. Wong, Generators and relations for classical groups[J], J. Algerbra 32 (1974).
[20] 张勤海，王俊新．子群为拟正规或自正规的有限群[J]，数学学报 1993．
[21] 张勤海．只具有半正规和反正规子群的有限群[J]，数学研究 1996．
[22] 李长稳，於遒，骆公志．可解群的一些充分条件[J]，淮阴师范学院学报 2004．
[23] 钟祥贵．有限群的某些sylow子群与可解性[J]，邵阳学院学报 2004．
[24] 李晓华．2-极大子群是次正规子群的有限群[J]，四川师范大学学报(自然科学版)2000．
[25] Zvonimir Janko, Elements of order at most 4 in finite 2-groups, Journal of Group Theory.
[26] Peter Schmid, On the Automorphism Group of Extraspecial 2-Groups, Journal of Algebra234, 492_506, 2000.
[27] Zvonimir Janko, A classification of finite 2-groups with exactly three involutions, Journal of Algebra 291 (2005) 505-533.
[28] Zvonirnir Janko, Finite 2-groups with a self-centralizing elementary abelian subgroup of order 8, Journal of Algebra 269 (2003) 189-214.
[29] M. F. Newman and Ming-Yao Xu, A classification of metacyclic p-groups, Research Report No, 32/1990.
[30] Yakov Berkovich, On Subgroups of Finite p-Groups, Journal of Algebra 224, 198]240_2000.
[31] Legovini P. Finite groups whose subgroups are either subnormal or pronormal Ⅱ[J], Rend Sem Math. Uni Padova, 1981.