Finite groups with $\Bbb P$-subnormal 2-maximal subgroups

TL;DR

This paper investigates finite groups in which every 2-maximal subgroup is $P$-subnormal, providing insights into their structure and subgroup properties.

Contribution

It characterizes finite groups where all 2-maximal subgroups are $P$-subnormal, a novel focus on subgroup chains with prime index.

Findings

Characterization of groups with all 2-maximal subgroups $P$-subnormal

Structural properties of such groups identified

Conditions under which subgroup chains exist clarified

Abstract

A subgroup $H$ of a group $G$ is called $\Bbb P$-{\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 $\Bbb P$-subnormal.