On the supersoluble hypercentre of a finite group. (English) Zbl 1436.20027
Let \(G\) be a finite group and \(H\leq G\). \(H\) is said to be \(S\)-permutable in \(G\) if \(H\) permutes with all Sylow subgroups of \(G\), and is said to be \(S\)-semipermutable in \(G\) if \(H\) permutes with all Sylow \(q\)-subgroups of \(G\) for the primes \(q\) not dividing \(|H|\). In this article, the authors give some sufficient conditions for a normal \(p\)-subgroup \(P\) of a finite group \(G\) to have every \(G\)-chief factor below it cyclic. They also prove the following theorem.
Theorem. Let \(P\in\mathrm{Syl}_{p}(G)\) and let \(d\) be a power of \(p\) such that \(1\leq d<|P|\). Assume that \(H\cap O^{p}(G)\) is \(S\)-semipermutable in \(G\) for all noncyclic subgroups \(H\) of \(P\) with \(|H|=d\). Then either \(|P\cap O^{p}(G)|>d\), or \(P\cap O^{p}(G)\) is cyclic, or else \(G\) is \(p\)-supersoluble.
Theorem. Let \(P\in\mathrm{Syl}_{p}(G)\) and let \(d\) be a power of \(p\) such that \(1\leq d<|P|\). Assume that \(H\cap O^{p}(G)\) is \(S\)-semipermutable in \(G\) for all noncyclic subgroups \(H\) of \(P\) with \(|H|=d\). Then either \(|P\cap O^{p}(G)|>d\), or \(P\cap O^{p}(G)\) is cyclic, or else \(G\) is \(p\)-supersoluble.
Reviewer: Haoran Yu (Changchun)
MSC:
| 20D20 | Sylow subgroups, Sylow properties, \(\pi\)-groups, \(\pi\)-structure |
| 20D10 | Finite solvable groups, theory of formations, Schunck classes, Fitting classes, \(\pi\)-length, ranks |
| 20D40 | Products of subgroups of abstract finite groups |
References:
| [1] | Ballester-Bolinches, A., Esteban-Romero, R., Asaad, M.: Products of finite groups. In: de Gruyter Expositions in Mathematics, vol. 53. Walter de Gruyter, Berlin (2010). doi:10.1515/9783110220612 · Zbl 1206.20019 |
| [2] | Ballester-Bolinches, A., Esteban-Romero, R., Qiao, S.H.: A note on a result of Guo and Isaacs about \[p\] p-supersolubility of finite groups. Arch. Math. (Basel) 106, 501-506 (2016). doi:10.1007/s00013-016-0901-7 · Zbl 1342.20020 · doi:10.1007/s00013-016-0901-7 |
| [3] | Berkovich, Y., Isaacs, I.M.: \[p\] p-Groups stabilizing certain subgroups. J. Algebra 414, 82-94 (2014). doi:10.1016/j.jalgebra.2014.04.026 · Zbl 1316.20012 |
| [4] | Chen, X., Guo, W., Skiba, A.N.: Some conditions under which a finite group belongs to a Baer-local formation. Commun. Algebra 42(10), 4188-4203 (2014). doi:10.1080/00927872.2013.806519 · Zbl 1316.20013 · doi:10.1080/00927872.2013.806519 |
| [5] | Doerk, K., Hawkes, T.: Finite soluble groups. In: De Gruyter Expositions in Mathematics, vol. 4. Walter de Gruyter, Berlin (1992). doi:10.1515/9783110870138 · Zbl 0753.20001 |
| [6] | Gorenstein, D.: Finite Groups. Chelsea, New York (1980) · Zbl 0463.20012 |
| [7] | Guo, Y., Isaacs, I.M.: Conditions on \[p\] p-supersolvability. Arch. Math. (Basel) 105, 215-222 (2015). doi:10.1007/s00013-015-0803-0 · Zbl 1344.20032 |
| [8] | Huppert, B.: Endliche Gruppen I. In: Grund. Math. Wiss., vol. 134. Springer, Berlin (1967) · Zbl 0217.07201 |
| [9] | Isaacs, I.M.: Semipermutable \[\pi\] π-subgroups. Arch. Math. (Basel) 102, 1-6 (2014). doi:10.1007/s00013-013-0604-2 · Zbl 1297.20018 · doi:10.1007/s00013-013-0604-2 |
| [10] | Li, Y., Qiao, S., Su, N., Wang, Y.: On weakly \[s\] s-semipermutable subgroups of finite groups. J. Algebra 371, 250-261 (2012). doi:10.1016/j.jalgebra.2012.06.025 · Zbl 1269.20020 |
| [11] | Su, N., Li, Y., Wang, Y.: The weakly \[s\] s-supplemented property of finite groups. Chin. J. Contemp. Math. 35(4), 1-12 (2014). doi:10.16205/j.cnki.cama.2015.0010 · Zbl 1316.20013 |
| [12] | Wang, L., Wang, Y.: On \[s\] s-subgroups of finite groups. Commun. Algebra 34, 143-149 (2006). doi:10.1080/00927870500346081 · Zbl 1087.20015 |
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