On the \(\mathcal X\Phi\)-hypercentre of finite groups.
Let \(G\) be a finite group. A subgroup \(H\) of a group \(G\) is said to be \(s\)-permutable in \(G\) if \(HP=PH\) for all Sylow subgroups \(P\) of \(G\). \textit{A. N. Skiba} [J. Algebra 315, No. 1, 192-209 (2007; Zbl 1130.20019)] defined \(H_{sG}\) to be the subgroup of \(H\) generated by all those subgroups of \(H\) which are \(s\)-permutable in \(G\) and called \(H\) weakly \(s\)-permutable in \(G\) if \(G\) has a subnormal subgroup \(T\) such that \(HT=G\) and \(T\cap H\) is a subgroup of \(H_{sG}\). Let \(\mathcal U\) be the class of all finite supersoluble groups. A chief factor \(H/K\) is said to be \(\mathcal U\)-central if the semidirect product of \(H/K\) by \(G/C_G(H/K)\) is supersoluble, and the product of all normal subgroups \(H\) of \(G\) such that all non-Frattini \(G\)-chief factors of \(H\) are \(\mathcal U\)-central is denoted by \(Z_{\mathcal U\Phi}(G)\). The main result (Theorem 1.4) of the present paper is as follows. Let \(X\leq E\) be normal subgroups of \(G\). Suppose that every non-cyclic Sylow subgroup \(P\) of \(X\) has a subgroup \(D\) such that \(1<|D|<|P|\) and every subgroup \(H\) of \(P\) with \(|H|=|D|\) and every cyclic subgroup of \(P\) of order 4 (if \(|D|=2\) and \(P\) is a non-Abelian 2-group) is weakly \(s\)-permutable in \(G\). If \(X\) is either \(E\) or \(F^*(E)\), the generalised Fitting subgroup of \(E\), then \(E\leq Z_{\mathcal U\Phi}(G)\).
scientific article; zbMATH DE number 5614702
scientific article