Finite groups with $\mathbb{P}$-subnormal primary cyclic 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 $H=H_0\subset H_1\subset ...\subset H_n=G$ such that $|H_i:H_{i-1}|$ is a prime for all $i$. In this paper, we study the groups in which all primary cyclic subgroups are $\mathbb{P}$-subnormal.