广义c~#-正规子群与有限群的可解性
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摘要
设G为有限群,H为G的子群.称H为G的广义c#-正规子群,如果存在G的正规子群K使得HK■G且H∩K是G的CAP-子群.该文利用某些2-极大子群、极大子群的Sylow子群或3-极大子群的广义c#-正规性,得到有限群可解的几个充分或充要条件.
Let G be a finite group and let H be a subgroup of G.His said to be a generalized c#-normal subgroup of Gif there exists a normal subgroup K of Gsuch that HK■G and H∩Kis a CAP-subgroup of G.By using the c#-normality of some 2-maximal subgroups,Sylow subgroups of maximal subgroups and 3-maximal subgroups,we obtain several sufficient or necessary conditions for a finite group to be solvable.
Let G be a finite group and let H be a subgroup of G.His said to be a generalized c#-normal subgroup of Gif there exists a normal subgroup K of Gsuch that HK■G and H∩Kis a CAP-subgroup of G.By using the c#-normality of some 2-maximal subgroups,Sylow subgroups of maximal subgroups and 3-maximal subgroups,we obtain several sufficient or necessary conditions for a finite group to be solvable.
引文
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[2]Doerk H,Hawkes T.Finite solvable groups[M].Berlin:Springer-Verlag,1992.
[3]Tomkinson M J.Cover-avoidance properties in finite solvable groups[J].Canad Math Bull,1976,19(2):213-216.
[4]Ezquerro L M.A contribution to the theory of finite supersolvable groups[J].Rend Sem Mat Univ Padova,1993,89:161-170.
[5]Guo X,Shum K P.Cover-avoidance properties and the structure of finite groups[J].J Pure Appl Algebra,2003,181:297-308.
[6]樊恽,郭秀云,岑嘉评.关于子群的两种广义正规性注记[J].数学年刊(A辑),2006,27(2):169-176.
[7]Wang Y.c-normality of groups and its properties[J].J Algebra,1996,180:954-965.
[8]韦华全.子群特性与有限群结构[D].广州:中山大学,2006.
[9]Wang Y,Wei H.c#-normality of groups and its properties[J].Algebr Represent Theory,2013,16(1):193-204.
[10]Wang Y,Wei H.On CAS-subgroups of finite groups[J].Israel J Math,2007,159:175-188.
[11]Robinson D J S.A course in the theory of groups[M].New York:Springer-Verlag,1980.