Semi cover-avoiding properties of finite groups. (English) Zbl 1223.20013
A subgroup \(H\) of \(G\) is said to cover \(M/N\) if \(HM=HN\). On the other hand, if \(H\cap M=H\cap N\), then \(H\) is said to avoid \(M/N\). (1) The subgroup \(H\) is said to have the cover-avoiding property in \(G\) if for every chief factor \(M/N\) of \(G\), \(H\) either covers \(M/N\) or avoids \(M/N\). (2) The subgroup \(H\) is said to have the semi cover-avoiding property in \(G\) if there is a fixed chief series \(1=G_0<G_1<\cdots<G_t=G\) of \(G\) such that, for every \(j=1,2,\dots,t\), \(H\) either covers \(G_j/G_{j-1}\) or avoids \(G_j/G_{j-1}\). In this paper are obtained characterizations of some group classes by the semi cover-avoiding property of some minimal subgroups.
Reviewer: Francisco Pérez-Monasor (Valencia)
MSC:
| 20D30 | Series and lattices of subgroups |
| 20D10 | Finite solvable groups, theory of formations, Schunck classes, Fitting classes, \(\pi\)-length, ranks |
| 20D15 | Finite nilpotent groups, \(p\)-groups |
| 20D40 | Products of subgroups of abstract finite groups |
Keywords:
finite groups; semi cover-avoiding property; supersolvable groups; chief series; nilpotent groups; formationsReferences:
| [1] | Ballester-Bolinches A. -normalizers and local definitions of saturated formations of finite groups. Israel J Math, 1989, 67(3): 312–326 · Zbl 0689.20036 · doi:10.1007/BF02764949 |
| [2] | Ballester-Bolinches A, Pedraza Aguilera M C. On minimal subgroups of finite groups. Acta Math Hungar, 1996, 73(4): 335–342 · Zbl 0930.20021 · doi:10.1007/BF00052909 |
| [3] | Doerk K, Hawkes T. Finite Soluble Groups. Berlin: Walter de Gruyter, 1992 · Zbl 0753.20001 |
| [4] | Fan Yun, Guo Xiuyun, Shum K P. Remarks on two generalizations of normality of Subgroups. Chinese Annals of Math (Chinese Series), 2006, 27A(2): 169–176 · Zbl 1109.20017 |
| [5] | Gaschütz W. Praefrattini gruppen. Arch Math, 1962, 13: 418–426 · Zbl 0109.01403 · doi:10.1007/BF01650090 |
| [6] | Guo Xiuyun, Guo Pengfei, Shum K P. On semi cover-avoiding subgroups of finite groups. J Pure and Appl Algebra, 2007, 209: 151–158 · Zbl 1117.20013 · doi:10.1016/j.jpaa.2006.05.027 |
| [7] | Guo Xiuyun, Shum K P. Cover-avoidance properties and the structure of finite groups. J Pure and Appl Algebra, 2003, 181: 297–308 · Zbl 1028.20014 · doi:10.1016/S0022-4049(02)00327-4 |
| [8] | Guo Xiuyun, Wang Lili. On finite groups with some semi cover-avoiding subgroups. Acta Math Sinica (English Series), 2007, 23: 1689–1696 · Zbl 1138.20016 · doi:10.1007/s10114-007-0946-4 |
| [9] | Huppert B. Endliche Gruppen I. New York: Springer, 1967 · Zbl 0217.07201 |
| [10] | Li Yangming, Wang Yanming, Wei Huaquan. On p-nilpotency of finite groups with some subgroups {\(\pi\)}-quasinormally embedded. Acta Math Hungar, 2005, 108(4): 283–298 · Zbl 1094.20007 · doi:10.1007/s10474-005-0225-8 |
| [11] | Robinson D J S. A Course in the Theory of Groups. 2nd ed. New York: Springer, 1996 |
| [12] | Yang Yuanwei, Li Xianhua. Semi CAP-subgroups and the properties of finite groups. Journal of Suzhou University (Natural Science Edition), 2008, 24(2): 1–6 · Zbl 1174.20004 |
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