On weakly \(s\)-permutably embedded subgroups of finite groups. II. (Q2907020)

In the last years many authors have analyzed the influence of various subgroup embedding properties on the structure of a finite group. Recall that a subgroup \(H\) of a finite group \(G\) is said to be \(s\)-permutably embedded in \(G\) if every Sylow subgroup of \(H\) is a Sylow subgroup of an \(s\)-permutable subgroup of \(G\) (i.e. a subgroup which permutes with every Sylow subgroup) [see \textit{A. Ballester-Bolinches} and \textit{M. C. Pedraza-Aguilera}, J. Pure Appl. Algebra 127, No. 2, 113-118 (1998; Zbl 0928.20020)].NEWLINENEWLINE In this paper, the authors deal with the following embedding property: A subgroup \(H\) is called weakly \(s\)-permutably embedded in \(G\) if there exist a subnormal subgroup \(T\) of \(G\) and an \(s\)-permutably embedded subgroup \(H_{se}\) of \(G\) contained in \(H\) such that \(G=HT\) and \(H\cap T\leq H_{se}\) [see \textit{A. N. Skiba}, J. Algebra 315, No. 1, 192-209 (2007; Zbl 1130.20019)].NEWLINENEWLINE The main theorem is the following: Theorem 1.6. Let \(\mathcal F\) be a saturated formation containing \(\mathcal U\), the class of all supersoluble groups, and let \(G\) be a finite group which has a normal subgroup \(E\) such that \(G/E\in\mathcal F\). Assume that every noncyclic Sylow subgroup \(P\) of \(F^*(E)\), the generalized Fitting subgroup of \(E\), satisfies the following property: \((\Delta_1)\) Suppose that \(P\) has a subgroup \(D\) such that \(1<|D|<|P|\) and all subgroups \(H\) of \(P\) with order \(|H|=|D|\) are \(s\)-permutably embedded in \(G\). Also, when \(p=2\) and \(|D|=2\), assume that each cyclic subgroup of \(P\) of order four is weakly \(s\)-permutably embedded in \(G\). -- Then \(G\in\mathcal F\).NEWLINENEWLINE For part I see the authors and \textit{Y.-M. Li}, Commun. Algebra 37, No. 3, 1086-1097 (2009; Zbl 1177.20036).