On the partial $ \mathscr L $-$ \Pi $-property of subgroups of finite groups

TL;DR

This paper investigates the structure of finite groups where certain prime power order subgroups satisfy a specific partial $\mathscr L$-$\Pi$-property, extending understanding of subgroup influence on group structure.

Contribution

It introduces the concept of the partial $\mathscr L$-$\Pi$-property for subgroups and explores its implications on the structure of finite groups.

Findings

Subgroups of prime power order satisfying the property influence group structure.

Characterization of finite groups with such subgroups.

New conditions under which groups exhibit certain structural properties.

Abstract

Let $ H $ be a subgroup of a finite group $ G $. We say that $ H $ satisfies the partial $ \mathscr L $-$ \Pi $-property in $ G $ if $ H\unlhd G $, or if $ | G / K : \mathrm{N} _{G / K} (HK/K)| $ is a $ \pi (HK/K) $-number for any $ G $-chief factor of type $ H^{G}/K $ with $ H_{G}\leq K $. In this paper, we investigate the structure of finite groups under the assumption that some subgroups of prime power order satisfy the partial $ \mathscr L $-$ \Pi $-property.