On finite groups with some primary subgroups satisfying partial $S$-$\Pi$-property

TL;DR

This paper investigates the structure of finite groups where certain primary subgroups satisfy a specific partial $S$-$ ext{ extPi}$-property related to chief series and Sylow subgroups.

Contribution

It introduces and studies the properties of primary subgroups satisfying partial $S$-$ ext{ extPi}$-property in finite groups, expanding understanding of their structural implications.

Findings

Characterization of finite groups with primary subgroups satisfying partial $S$-$ ext{ extPi}$-property.

Conditions under which such subgroups influence group structure.

New criteria for subgroup properties affecting group composition.

Abstract

A $p$-subgroup $H$ of a finite group $G$ is said to satisfy partial $S$-$\Pi$-property in $G$ if $G$ has a chief series $\Gamma_{G}: 1=G_{0}<G_{1}<\cdots<G_{n}=G$ such that for every $G$-chief factor $G_{i}/G_{i-1}$ $(1\leqslant i\leqslant n)$ of $\Gamma_{G}$, either $(H\cap G_{i})G_{i-1}/G_{i-1}$ is a Sylow $p$-subgroup of $G_{i}/G_{i-1}$ or $|G/G_{i-1}: N_{G/G_{i-1}}((H\cap G_{i})G_{i-1}/G_{i-1})|$ is a $p$-number. In this paper, we mainly investigate the structure of finite groups with some primary subgroups satisfying partial $S$-$\Pi$-property.