On \(SS\)-semipermutable subgroups of finite groups. (Q719788)

Let \(G\) be a finite group and \(H\) a subgroup of \(G\). \textit{Q. Zhang} and \textit{L. Wang} [Acta Math. Sin. 48, No. 1, 81-88 (2005; Zbl 1119.20026)] introduced the concept of \(S\)-semipermutability: \(H\) is said to be \(S\)-semipermutable in \(G\) if \(HG_p=G_pH\) for every Sylow \(p\)-subgroup \(G_p\) of \(G\) with \(\gcd(p,|H|)=1\). In the present paper, the concept of \(SS\)-semipermutability is introduced: \(H\) is said to be \(SS\)-semipermutable in \(G\) if there exists a normal subgroup \(T\) of \(G\) and an \(S\)-semipermutable subgroup \(H_S\) of \(G\) contained in \(H\) such that \(G=HT\) and \(H\cap T\leq H_S\). The author provides the following extension of a result by \textit{A. N. Skiba} [J. Algebra 315, No. 1, 192-209 (2007; Zbl 1130.20019)]. Let \(\mathcal F\) be a saturated formation containing the class of supersoluble groups and \(E\) a normal subgroup of \(G\) with \(G/E\in\mathcal F\), and suppose that every non-cyclic Sylow subgroup \(P\) of \(F^*(E)\), the generalised Fitting subgroup of \(E\), has a subgroup \(D\) such that \(1<|D|<|P|\) and every subgroup of \(P\) with order \(|D|\) or with order \(2|D|\) (if \(P\) is a non-Abelian 2-group and \([P:D]>2\)) is \(SS\)-semipermutable. Then \(G\in\mathcal F\) (Theorem 3.5).