Groups in which some primary subgroups are weakly \(s\)-supplemented
For a finite group \(G\), a subgroup is called \(s\)-permutable if it permutes with every Sylow subgroups of \(G\). The concept of \(s\)-permutable subgroups was generalised by \textit{A. Skiba} in [J. Algebra 315, 192--209 (2007; Zbl 1130.20019)] as follows.NEWLINENEWLINE If \(H\leq G\) is any subgroup, then let \(H_{sG}\) denote the subgroup generated by all \(s\)-permutable subgroups of \(G\) contained in \(H\). A subgroup \(H\leq G\) is called weakly \(s\)-supplemented in \(G\) if there is a subgroup \(T\leq G\) such that \(TH=G\) and \(T\cap H\leq H_{sG}\). If, moreover, such a \(T\) can be chosen to be subnormal in \(G\), then \(H\) is called weakly \(s\)-permutable in \(G\). In the aformentioned paper of Skiba it was proved that if \(\mathfrak F\) is a saturated formation containing all supersolvable groups and \(G/E\in\mathfrak F\) for some \(E\vartriangleleft G\) such that some specific subgroups of Sylow \(p\)-subgroups of \(E\) are weakly \(s\)-permutable in \(G\), then \(G\in \mathfrak F\) holds. This theorem provided a common generalisation of many previously known results.NEWLINENEWLINE In his paper, Skiba also raised the question, whether the assumption ``weakly \(s\)-permutable'' in his theorem can be weekened to ``weakly \(s\)-supplemented''. It turned out that the answer to this question is negative in general [\textit{W. Guo} et al., Sci. China, Ser. A 52, 2132--2144 (2009; Zbl 1193.20021)]. The main result of the paper under review is to give some additional conditions each of which guarantees a positive answer to the question of Skiba.
scientific article; zbMATH DE number 6656640
scientific article