Asaad, M.; Ballester-Bolinches, A.; Pedraza Aguilera, M. C. A note on minimal subgroups of finite groups. (English) Zbl 0856.20015 Commun. Algebra 24, No. 8, 2771-2776 (1996). Let \(G\) be a finite group and \(\psi(G)=\langle x\mid x\in G\), \(|x|\) is a prime number or \(|x|=4\rangle\). A subgroup \(H\) of a group \(G\) is \(\pi\)-quasinormal in \(G\) if \(H\) permutes with every Sylow subgroup of \(G\). An element \(x\in G\) is \(\pi\)-quasinormal in \(G\) if \(\langle x\rangle\) is a \(\pi\)-quasinormal subgroup of \(G\).Let \(\mathfrak F\) be a saturated formation containing the formation of all supersoluble groups. Let \(G\) be a group with a normal subgroup \(H\) such that \(G/H\in{\mathfrak F}\). It is proved that if every generator of \(\psi(H)\) is \(\pi\)-quasinormal in \(G\) or if \(H\) has a Sylow tower of supersoluble type and for each Sylow subgroup \(P\) of \(H\) every generator of \(\psi(P)\) is \(\pi\)-quasinormal in \(N_G(P)\), then \(G\in{\mathfrak F}\). Reviewer: N.M.Kurnosenko (Gomel) Cited in 3 ReviewsCited in 16 Documents MSC: 20D25 Special subgroups (Frattini, Fitting, etc.) 20D10 Finite solvable groups, theory of formations, Schunck classes, Fitting classes, \(\pi\)-length, ranks 20D20 Sylow subgroups, Sylow properties, \(\pi\)-groups, \(\pi\)-structure Keywords:finite groups; Sylow subgroups; \(\pi\)-quasinormal subgroups; saturated formations; supersoluble groups; Sylow towers PDFBibTeX XMLCite \textit{M. Asaad} et al., Commun. Algebra 24, No. 8, 2771--2776 (1996; Zbl 0856.20015) Full Text: DOI