\(\mathbb{P}\)-subnormal subgroups and the structure of finite groups (Q6054812)

Let \(G\) be a finite group and let \(\mathbb{P}\) be the set of all primes. A subgroup \(H\) of \(G\) is called \(\mathbb{P}\)-subnormal whenever \(H = G\) or there exists a chain of subgroups \(H=H_{0} \leq \cdots \leq H_{t}=G\) such that \(|H_{i} : H_{i-1}| \in \mathbb{P}\) for all \(i \in \{1, \ldots, t\}\). In the paper under review, the authors study finite groups in which some 2-maximal subgroups or cyclic subgroups are \(\mathbb{P}\)-subnormal subgroups. Some conditions for \(G\) to possess an ordered Sylow tower of supersolvable type are given.