Finite Groups with \(\mathbb{P}\)-Subnormal Schmidt Subgroups

A subgroup \(H\) $𝐻$ of a group \(G\) $𝐺$ is called \(\mathbb{P}\) $\mathbb{P}$ -subnormal in \(G\) $𝐺$ whenever either \(H=G\) $𝐻=𝐺$ or there is a chain of subgroups \(H=H_{0}\subset H_{1}\subset\mathinner{\ldotp\ldotp\ldotp}\subset H_{n}=G\) $(𝐻=\mathrm{subscript}(𝐻, 0))\land \subseteq \mathrm{subscript}(𝐻, 1)\land \subseteq \dots \land \subseteq \mathrm{subscript}(𝐻, 𝑛)\land (=𝐺)$ such that \(|H_{i}:H_{i-1}|\) $\mathrm{fragments}|\mathrm{subscript}(𝐻, 𝑖):\mathrm{subscript}(𝐻, 𝑖-1)|$ is a prime for every \(i=1,2,\mathinner{\ldotp\ldotp\ldotp},n\) $𝑖=\left[1, 2, \dots , 𝑛\right]$ . We study the structure of a finite group \(G\) $𝐺$ all of whose Schmidt subgroups are \(\mathbb{P}\) $\mathbb{P}$ -subnormal. The obtained results complement the answer to Problem 18.30 in the Kourovka Notebook..