摘要
令E是有限群G的一个正规子群,且U是所有有限超可解群的集合.E称为在G中是p-超循环嵌入的,如果E的每个pd-阶的G-主因子是循环的.G的子群H称为在G中是U-Φ-可补充的,如果存在G的一个次正规子群T,使得G=HT,且(H∩T)H_G/H_G≤Φ/(H/H_G)Z_U(G/H_G),其中Z_U(G/H_G)是商群G/H_G的U-超中心.作者证明,如果E的一些p-子群在G中是U-Φ-可补充的,那么E在G中是p-超循环嵌入的.作为应用,得到了有限群是p-超可解的若干判断准则,并且推广了一些已知的结果.
Let E be a normal subgroup of a finite group G and U the class of all finite super solvable groups. E is said to be p-hypercyclically embedded in G if every pd-G-chief factor below E is cyclic. A subgroup H of G is U-Φ-supplemented in G if there exists a subnormal subgroup T of G such that G =HT and(H∩T)H_G/H_G≤Φ(H/H_G)Z_U(G/H_G),where Z_U(G/H_G) is the U-hypercentre of G/H_G In this paper, it is proved that E is phypercyclically embedded in G if some classes of p-subgroups of E are U-Φ-supplemented in G. As applications, some new characterizations of p-supersolvability of finite groups are obtained and some recent results are extended.
引文
[1]Doerk K,Hawkes T.Finite soluble groups[M].Berlin,New York:Walter de Gruyter,1992.
[2]Guo W B.Structure theory for canonical classes of finite groups[M].Berlin,Heidelberg,Dordrecht,New York:Walter de Gruyter,2015.
[3]Huppert B.Endliche gruppen I[M].Berlin,New York:Springer-Verlag,1967.
[4]Guo W B,Skiba A N.On some classes of finite quasi-F-groups[J].J Group Theory,2009,12:407-417.
[5]Ballester-Bolinches A,Ezquerro L M,Skiba A N.Local embeddings of some families of subgroups of finite groups[J].Acta Math Sin(Engl Ser),2009,25:869-882.
[6]Guo W B,Skiba A N,Tang X Z.On boundary factors and traces of subgroups of finite groups[J].Commun Math Stat,2014,2:349-361.
[7]Huo L J,Guo W B,Makhnev A A.On nearly SS-embedded subgroups of finite groups[J].Chinese Annals of Mathematics,Series B,2014,35:885-894.
[8]Skiba A N.On two questions of L A Shemetkov concerning hypercyclically embedded subgroups of finite groups[J].J Group Theory,2010,13:841-850.
[9]Skiba A N.Cyclicity conditions for G-chief factors of a normal subgroup of a group G(Russian)[J].Sibirsk Mat Zh,2011,52:161-166.
[10]Skiba A N.A characterization of the hypercyclically embedded subgroups of finite groups[J].J Pure Appl Algebra,2011,215:257-261.
[11]Su N,Li Y Y,Wang Y Y.A criterion of p-hypercyclically embedded subgroups of finite groups[J].J Algebra,2014,400:82-93.
[12]Wang Y Y.c-Normality of groups and its properties[J].J Algebra,1996,180:945-965.
[13]Alsheik Ahmad A Y,Jaraden J J,Skiba A N.On u_c-normal subgroups of finite groups[J].Algebra Colloq,2007,14:25-36.
[14]Li X H,Zhao T.SΦ-supplemented subgroups of finite groups[J].Ukrainian Math J,2012,64:102-109.
[15]Ballester-Bolinches A,Esteban-Romero R,Asaad M.Products of finite groups[M].Berlin,New York:Walter de Gruyter,2010.
[16]Ballester-Bolinches A,Ezquerro L M,Skiba A N.On second maximal subgroups of Sylow subgroups of finite groups[J].J Pure Appl Algebra,2011,215:705-714.
[17]Guo W B,Skiba A N.On FΦ~*-hypercentral subgroups of finite groups[J].J Algebra,2012,372:275-292.
[18]Li B J,Guo W B.On some open problems related to X-permutability of subgroups[J].Comm Algebra,2011,39:757-771.
[19]Gagen T M.Topics in finite groups[M].New York,Melbourne:Cambridge,1976.
[20]Schmid P.Subgroups permutable with all Sylow subgroups[J].J Algebra,1998,207:285-293.
[21]Shemetkov L A,Skiba A N.On the XΦ-hypercentre of finite groups[J].J Algebra,2009,322:2106-2117.
[22]Ballester-Bolinches A,Pedraza-Aguilera M C.On minimal subgroups of finite groups[J].Acta Math Hungar,1996,73:335-342.
[23]Guo W B.The theory of classes of groups[M].Beijing,New York,Dordrecht,Boston,London:Science Press-Kluwer Academic Publishers,2000.
[24]Robinson D J S.A course in the theory of groups[M].Berlin,New York:SpringerVerlag,1982.
[25]Ballester-Bolinches A,Guo X Y.On complemented subgroups of finite groups[J].Arch Math,1992,72:161-166.
[26]Buckley J.Finite groups whose minimal subgroups are normal[J].Math Z,1970,116:15-17.
[27]Ramadan M.Influence of normality on maximal subgroups of Sylow subgroups of a finite groups[J].Acta Math Hungar,1992,59(1-2):107-110.
