Finite groups with $\mathbb{P}$-subnormal and strongly permutable subgroups

TL;DR

This paper investigates finite groups where certain subgroups, specifically $ ext{P}$-subnormal and strongly permutable subgroups, influence the group's structure, proving that groups with all strongly permutable primary cyclic subgroups are supersoluble.

Contribution

It introduces the concepts of $ ext{P}$-subnormal and strongly permutable subgroups and characterizes the structure of groups with these properties, especially proving supersolubility.

Findings

Groups with all strongly permutable primary cyclic subgroups are supersoluble.

Characterization of groups with $ ext{P}$-subnormal Sylow subgroups.

Analysis of the permutizer subgroup properties in finite groups.

Abstract

Let $H$ be a subgroup of a group $G$. The permutizer $P_G(H)$ is the subgroup generated by all cyclic subgroups of $G$ which permute with $H$. A subgroup $H$ of a group $G$ is strongly permutable in $G$ if $P_U(H)=U$ for every subgroup $U$ of $G$ such that~$H\le U\le G$. We investigate groups with $\mathbb{P}$-subnormal or strongly permutable Sylow and primary cyclic subgroups. In particular, we prove that groups with all strongly permutable primary cyclic subgroups are supersoluble.