A result on s-semipermutable subgroups of finite groups and some applications

TL;DR

This paper investigates conditions under which a finite group is p-supersolvable, focusing on the properties of certain subgroups related to Sylow p-subgroups, and provides new proofs and generalizations of existing results.

Contribution

It establishes new criteria involving s-semipermutable subgroups that ensure p-supersolvability in finite groups, simplifying and extending previous theorems.

Findings

G is p-supersolvable if N_G(P) is p-supersolvable and a specific subgroup H is s-semipermutable

Provides simplified proofs for existing results in group theory

Generalizes known results about p-supersolvability and subgroup properties

Abstract

Let $p$ be a prime number, $G$ be a $p$-solvable finite group and $P$ be a Sylow $p$-subgroup of $G$. We prove that $G$ is $p$-supersolvable if $N_G(P)$ is $p$-supersolvable and if there is a subgroup $H$ of $P$ with $P' \le H \le \Phi(P)$ such that $H$ is $s$-semipermutable in $G$. As applications, we simplify the proofs of some known results and also generalize some known results.