A note on weakly \(\sigma\)-\(n\)-embedded subgroups of \(\sigma\)-full groups (Q6635338)

Let \(G\) be a finite group and let \(\sigma=\{ \sigma_{i} \mid i \in I \}\) be a partition of the set of all primes. A subgroup \(H \leq G\) is \(\sigma\)-subnormal in \(G\) if there is a chain of subgroups \(H=H_{0} \leq H_{1} \leq \dots \leq H_{n}=G\) where for every \(j=1, \ldots n\), \(H_{j-1}\) is normal in \(H_{j}\) or \(H_{j}/(H_{j-1})^{H_{j}}\) is a \(\sigma_{i}\)-group for some \(i \in I\). In addition, \(H\) is weakly \(\sigma\)-\(n\)-embedded in \(G\), if there is a \(\sigma\)-subnormal subgroup \(T\leq G\) such that \(H^{G}=HT\) and \(H \cap T \leq H_{\sigma G}\), where the subgroup \(H_{\sigma G}\) is generated by all \(\sigma\)-subnormal subgroups of \(G\) contained in \(H\).\N\NThe paper under review is devoted to the study of weakly \(\sigma\)-\(n\)-embedded subgroups groups and their properties.