Finite groups with \(\mathbb{P} \)-subnormal Schmidt subgroups (Q6597572)

A finite group \(G\) is a Schmidt group if \(G\) is non-nilpotent but all its proper subgroups are nilpotent. A subgroup \(H\) of \(G\) is \(\mathbb{P}\)-subnormal whenever either \(H=G\) or there is a chain of subgroups \(H=H_{0} \leq H_{1} \leq \ldots \leq H_{n}=G\) such that \(|H_{i} : H_{i-1}|\) is a prime for every \(i=1,2,\ldots, n\). In [\textit{V. N. Tyutyanov}, Probl. Fiz. Mat. Tekh. 2015, No. 1(22), 88--91 (2015; Zbl 1326.20020)] it is proved that a finite group is soluble if all its Schmidt subgroups are \(\mathbb{P}\)-subnormal, by answering in the affirmative Problem 18.30 in [\textit{V. D. Mazurov} (ed.) and \textit{E. I. Khukhro} (ed.), The Kourovka notebook. Unsolved problems in group theory. 18th edition. Novosibirsk: Institute of Mathematics, Russian Academy of Sciences, Siberian Div. (2014; Zbl 1372.20001)].\N\NIn the paper under review, the authors study the structure of finite groups all of whose Schmidt subgroups are \(\mathbb{P}\)-subnormal. The results obtained complement the answer to Problem 18.30 quoted above.