A note on the $ \Pi $-property of some subgroups of finite groups

TL;DR

This paper investigates the $ \Pi $-property of subgroups in finite groups, providing criteria for $ p $-supersolubility and $ p $-nilpotency, and extending existing results on subgroup properties.

Contribution

It introduces new criteria for $ p $-supersolubility and $ p $-nilpotency based on the $ \Pi $-property, extending previous subgroup property results.

Findings

Criteria for $ p $-supersolubility of finite groups.

Criteria for $ p $-nilpotency of finite groups.

Extension of known results on subgroup $ \Pi $-property.

Abstract

Let $ H $ be a subgroup of a finite group $ G $. We say that $ H $ satisfies the $ \Pi $-property in $ G $ if for any 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. In this paper, we obtain some criteria for the $ p $-supersolubility or $ p $-nilpotency of a finite group and extend some known results by concerning some subgroups that satisfy the $ \Pi $-property.