On $\Pi$-supplemented subgroups of a finite group

TL;DR

This paper studies the structure of finite groups where certain primary subgroups are $ ext{Pi}$-supplemented, extending previous results by analyzing subgroup properties related to chief factors and normalizers.

Contribution

It introduces the concept of $ ext{Pi}$-supplemented subgroups and proves new structural results for finite groups under this assumption, improving earlier theorems.

Findings

Characterization of finite groups with $ ext{Pi}$-supplemented primary subgroups

Generalization of previous subgroup structure theorems

Enhanced understanding of subgroup interactions in finite groups

Abstract

A subgroup $H$ of a finite group $G$ is said to satisfy $\Pi$-property in $G$ if for every chief factor $L/K$ of $G$, $|G/K:N_{G/K}(HK/K\cap L/K)|$ is a $\pi(HK/K\cap L/K)$-number. A subgroup $H$ of $G$ is called to be $\Pi$-supplemented in $G$ if there exists a subgroup $T$ of $G$ such that $G=HT$ and $H\cap T\leq I\leq H$, where $I$ satisfies $\Pi$-property in $G$. In this paper, we investigate the structure of a finite group $G$ under the assumption that some primary subgroups of $G$ are $\Pi$-supplemented in $G$. The main result we proved improves a large number of earlier results.