半正规、C-正规对群结构的影响
|
摘要
称群G的子群A为G的半正规子群，如果存在一个子群B使得AB=G，且对B的任意真子群B_1，AB_1是G的真子群；而称H为G的C-正规子群，若存在K G使得G=HK且H∩K≤Corec(H)．
本文结合有限群G的某些特殊子群(如，极小子群，极大子群，Sylow子群及Sylow子群的极大子群等)的“半正规或C-正规性”来讨论有限群的可解性，超可解性及幂零性，得到了有限群可解，超可解及幂零的若干充分或充要条件，同时推广了某些著名结果．特别地，引进群G的两个特征子群U(G)及V(G)，用这两个子群来刻划有限群的结构，得到了有限群超可解，幂零的一些充分条件，减弱了某些已知定理的条件．
以下是本文的主要结果：
1．有限群G可解当且仅当G的每一极大子群在G中或半正规或C-正规．
2．设G为有限群．若G′的每一极小子群在G中C-正规，则G可解．
3．若群G的每个Sylow子群的极大子群在G中或半正规或C-正规，则G超可解．
4．设N G，G／N超可解．若N之极小子群及2~2阶循环子群在G中或半正规或C-正规，则G超可解．
5．设G是有限群，N G，G／N超可解．若N之极小子群含于U(G)，且2~2阶循环子群在G中或半正规或C-正规，则G超可解．
6．设G是有限群，N G，G／N幂零．若N之极小子群含于V(G)，且2~2阶循环子群在G中或半正规或C-正规，则G是幂零群．
A subgroup A is called semi-normal in G if there exists a subgroup B of G such that G=AB, and AB1 < G for every proper subgroup B1 of B. A subgroup H is called C-normal in G if there exists a normal subgroup K of G such that G =HK and H K We investigate the influence of some special subgroups (such as minimal subgroups, maximal subgroups, Sylow subgroups, maximal subgroups of Sylow subgroups) on the structure of a finite group G by combining the concept of semi-normal subgroup and C-normal subgroup, and we get some sufficient or necessary conditions for finite groups to be solvable, supersolvable, nilpotent. Our theorems generalize some previously known results. At the same time, we get some new characterization of supersolvable and nilpotent groups by introduce the new concepts of U(G) and V(G) .
The following theorems are some of the main results in this thesis.
1. A finite group G is solvable if and only if every maximal subgroup of G is semi-normal or C-normal in G.
2. Let G be a finite group. Suppose that every minimal subgroup in G' is C-normal in G, then G is solvable.
3. Let G be a finite group. Suppose P1 is semi-normal or C-normal in G for every Sylow subgroup P of G and every maximal subgroup P1 of P, then G is supersolvable.
4. Let N< G and G/N is supersolvable. Suppose that (x) is semi-normal or C-normal in G for every element x of N with prime order or order 4, then G is supersolvable.
5. Let N 6. Let N < G and G/N is nilpotent. Suppose that (x) is contain in V(G) for every element x of N with prime order, and suppose that (y) is semi-normal or C-normal in G for every element y of N with order 4, then G is nilpotent.
本文结合有限群G的某些特殊子群(如，极小子群，极大子群，Sylow子群及Sylow子群的极大子群等)的“半正规或C-正规性”来讨论有限群的可解性，超可解性及幂零性，得到了有限群可解，超可解及幂零的若干充分或充要条件，同时推广了某些著名结果．特别地，引进群G的两个特征子群U(G)及V(G)，用这两个子群来刻划有限群的结构，得到了有限群超可解，幂零的一些充分条件，减弱了某些已知定理的条件．
以下是本文的主要结果：
1．有限群G可解当且仅当G的每一极大子群在G中或半正规或C-正规．
2．设G为有限群．若G′的每一极小子群在G中C-正规，则G可解．
3．若群G的每个Sylow子群的极大子群在G中或半正规或C-正规，则G超可解．
4．设N G，G／N超可解．若N之极小子群及2~2阶循环子群在G中或半正规或C-正规，则G超可解．
5．设G是有限群，N G，G／N超可解．若N之极小子群含于U(G)，且2~2阶循环子群在G中或半正规或C-正规，则G超可解．
6．设G是有限群，N G，G／N幂零．若N之极小子群含于V(G)，且2~2阶循环子群在G中或半正规或C-正规，则G是幂零群．
A subgroup A is called semi-normal in G if there exists a subgroup B of G such that G=AB, and AB1 < G for every proper subgroup B1 of B. A subgroup H is called C-normal in G if there exists a normal subgroup K of G such that G =HK and H K
The following theorems are some of the main results in this thesis.
1. A finite group G is solvable if and only if every maximal subgroup of G is semi-normal or C-normal in G.
2. Let G be a finite group. Suppose that every minimal subgroup in G' is C-normal in G, then G is solvable.
3. Let G be a finite group. Suppose P1 is semi-normal or C-normal in G for every Sylow subgroup P of G and every maximal subgroup P1 of P, then G is supersolvable.
4. Let N< G and G/N is supersolvable. Suppose that (x) is semi-normal or C-normal in G for every element x of N with prime order or order 4, then G is supersolvable.
5. Let N
引文
[1]苏向盈．有限群的半正规子群．数学杂志，1988，8(1)：5-9．
[2]Wang Y. G-Normality of Groups and Its Properties[J]. J. of Algebra,1996,180: 954-965.
[3]张远达．有限群构造(上，下)．科学出版社，1982．
[4]王品超．超可解的若干充分条件．数学学报，1990，33(4)：480-484．
[5]徐明曜等．有限群导引(上，下)．北京：科学出版社，1999．
[6]郭秀云．正规p-补与幂零群的特征．数学杂志．1987，7(4)：403-406．
[7]Rose J S. On finite insoluble groups with nilpotent maximal subgroups. J. Algebra,1977,48: 182-196.
[8][德]贝·胡佩特．有限群论(中译本第一卷)，福建人民出版社，1992．
[9]陈重穆．内外-∑群与极小非∑群．重庆：西南师范大学出版社，1988．
[10]王燕鸣．极小子群对有限群结构的影响．数学学报，2001，44(2)：197-200．
[11]钟祥贵．极小子群与少幂零性．广西科学，1999，6(4)：243-245．
[12]温凤桐，管延勇．关于半正规子群的几个定理．山东建材学院学报，2001，15(1)，66-68．
[13]王品超，温凤桐，李文祥．有限群的C-正规子群．曲阜师范大学学报，1997，23(4)：5-7．
[14]李样明，C-正规与有限群的结构．工程数学学报，2001，18(4)：88-92．
[15]王坤仁．Sylow子群皆半正规的有限群．四川师范大学学报(自然科学版)，1995，18(3)：1-4．
[2]Wang Y. G-Normality of Groups and Its Properties[J]. J. of Algebra,1996,180: 954-965.
[3]张远达．有限群构造(上，下)．科学出版社，1982．
[4]王品超．超可解的若干充分条件．数学学报，1990，33(4)：480-484．
[5]徐明曜等．有限群导引(上，下)．北京：科学出版社，1999．
[6]郭秀云．正规p-补与幂零群的特征．数学杂志．1987，7(4)：403-406．
[7]Rose J S. On finite insoluble groups with nilpotent maximal subgroups. J. Algebra,1977,48: 182-196.
[8][德]贝·胡佩特．有限群论(中译本第一卷)，福建人民出版社，1992．
[9]陈重穆．内外-∑群与极小非∑群．重庆：西南师范大学出版社，1988．
[10]王燕鸣．极小子群对有限群结构的影响．数学学报，2001，44(2)：197-200．
[11]钟祥贵．极小子群与少幂零性．广西科学，1999，6(4)：243-245．
[12]温凤桐，管延勇．关于半正规子群的几个定理．山东建材学院学报，2001，15(1)，66-68．
[13]王品超，温凤桐，李文祥．有限群的C-正规子群．曲阜师范大学学报，1997，23(4)：5-7．
[14]李样明，C-正规与有限群的结构．工程数学学报，2001，18(4)：88-92．
[15]王坤仁．Sylow子群皆半正规的有限群．四川师范大学学报(自然科学版)，1995，18(3)：1-4．