On some aspects of the Kegel-Wielandt \(\sigma\)-problem (Q6635336)

Let \(G\) be a finite group and \(\pi \in \pi(G)\). A subgroup \(H \leq G\) is \(p\)-subnormal if the intersections of \(H\) with any Sylow \(p\)-subgroups of \(G\) are Sylow \(p\)-subgroups \(H\). Let \(\sigma=\{\sigma_{i} \mid i \in I \}\) be a partition of the set of all primes. The subgroup \(H\) is \(\sigma\)-subnormal in \(G\) if there exists a chain of subgroups \(H=H_{0} \leq H_{1} \leq \dots \leq H_{n}=G\) such that for every \(i = 1, \ldots, n\) either \(H_{j-1}\) is normal in \(H_{j}\) or \(H_{j}/(H_{j-1})_{H_{j}}\) is a \(\sigma_{i}\)-subgroup for some \(i \in I\). \textit{O. H. Kegel} [Math. Z. 78, 205--221 (1962; Zbl 0102.26802)] formulated the following conjecture: \(H\) is subnormal in \(G\) if and only if \(H\) is \(p\)-subnormal in \(G\) for all primes \(p \in \pi(G)\).\N\NThe authors characterize the class of all finite groups in which all \(\sigma_{i}\)-subnormal subgroups form a join-semilattice for any \(j \in I\). Furthermore, for such groups, the authors investigate the following Kegel-Wielandt \(\sigma\)-problem: \(H\) is \(\sigma\)-subnormal in \(G\) if \(H\) is \(\sigma_{i}\)-subnormal in \(G\) for all \(i \in I\).