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

Abstract

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