Finite groups with $\mathbb{P}$-subnormal 2-maximal subgroups

Abstract

A subgroup $H$ of a group $G$ is called $\mathbb{P}$-{\sl subnormal} in $G$ if either $H=G$ or there is a chain of subgroups $H=H_0\subset H_1\subset ...\subset H_n=G$ such that $|H_i:H_{i-1}|$ is prime for $1\le i\le n$. In this paper we study the groups all of whose 2-maximal subgroups are $\mathbb{P}$-subnormal.