On \(\mathfrak Z\)-permutability of minimal subgroups of finite groups.
Many authors have studied finite groups \(G\), under the assumption that all minimal subgroups of \(G\) are well-situated in the group. First results in this line are due to \textit{N. Itô} [Nagoya Math. J. 9, 123-127 (1955; Zbl 0066.01401)] and \textit{J. Buckley} [Math. Z. 116, 15-17 (1970; Zbl 0202.02303)]. \textit{A. Yokoyama} [Arch. Math. 26, 123-130 (1975; Zbl 0307.20012), ibid. 27, 572-575 (1976; Zbl 0348.20016)] and \textit{R. Laue} [J. Algebra 52, 347-353 (1978; Zbl 0383.20015)] extended these results in the framework of formation theory. Later the reviewer and \textit{M. C. Pedraza-Aguilera} [Acta Math. Hung. 73, No. 4, 335-342 (1996; Zbl 0930.20021)] and \textit{M. Asaad}, the reviewer and \textit{M. C. Pedraza-Aguilera} [Commun. Algebra 24, No. 8, 2771-2776 (1996; Zbl 0856.20015)] analysed the structure of finite groups with minimal subgroups (or subgroups of order \(4\)) quasinormal or \(S\)-quasinormal in the group. The present paper represents an attempt to extend and improve the results of the above mentioned authors by dealing with another interesting embedding property: \(\mathcal Z\)-permutability. A subgroup of a group \(G\) is said to be \(\mathcal Z\)-permutable in \(G\) if it permutes with every member of \(\mathcal Z\), here \(\mathcal Z\) means a complete set of Sylow subgroups of \(G\). The main result in the paper states that if \(\mathcal F\) is a saturated formation containing the class of supersoluble groups and \(\mathcal Z\) is a complete set of Sylow subgroups of \(G\) then \(G\) is an \(\mathcal F\)-group if and only if there is a normal subgroup \(H\) of \(G\) such that \(G/H\in{\mathcal F}\) and the cyclic subgroups of \(G_p\cap F^*(G)\) of prime order or order \(4\) (if \(p=2\)) are \(\mathcal Z\)-permutable in \(G\), for all \(G_p\in{\mathcal Z}\), where \(F^*(G)\) is the generalized Fitting subgroup of \(G\). We remark that in the proof of this theorem the authors use the classification of finite simple groups.
scientific article; zbMATH DE number 2103660