Finite groups with $\mathbb P$-subnormal primary cyclic subgroups

TL;DR

This paper investigates finite groups where every primary cyclic subgroup is $ ext{P}$-subnormal, providing insights into their subgroup structure and classification.

Contribution

It characterizes finite groups with all primary cyclic subgroups $ ext{P}$-subnormal, advancing understanding of subgroup chains and group structure.

Findings

All primary cyclic subgroups are $ ext{P}$-subnormal in the studied groups.

Classification results for groups with this property.

Structural properties related to subgroup chains.

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.