Structure of finite groups with some weakly $S$-semipermutable subgroups

Abstract

Let $ G $ be a finite group. If $ A\leq G $, recall that $ A $ is weakly $S$-semipermutable in $G$ provided there is $K\unlhd G$ such that $KA$ is $S$-permutable in $G$, and $K\cap A$ is $S$-semipermutable in $G$. The purpose of this paper is to demonstrate that weakly $S$-semipermutability of special types of subgroups in a finite group $ G $ can help us to determine structural properties of $ G $. For example, given a prime $p$, a $p$-soluble finite group $G$ and a Sylow $p$-subgroup $G_{p}$ of $G$, we will show that $G$ is $p$-supersoluble if the maximal subgroups of $G_{p}$ are weakly $S$-semipermutable in $ G$. Moreover, we use the concept of weakly $S$-semipermutability to prove new criteria for $p$-nilpotency of finite groups.

Keywords

• p-nilpotent
• p-supersoluble
• weakly s-semipermutable