On p-nilpotency of finite groups?

TL;DR

This paper introduces new criteria for determining p-nilpotency in finite groups based on properties of certain subgroups satisfying the $ ext{Pi}$-property and $ ext{Pi}$-normality, expanding understanding of group structure.

Contribution

It develops novel criteria for p-nilpotency in finite groups using subgroup properties related to the $ ext{Pi}$-property and $ ext{Pi}$-normality.

Findings

New criteria for p-nilpotency established

Criteria based on subgroup $ ext{Pi}$-property and $ ext{Pi}$-normality

Enhanced understanding of subgroup influence on group structure

Abstract

Let $H$ be a subgroup of a group $G$. $H$ is said satisfying $\Pi$-property in $G$, if $|G/K:N_{G/K}(HK/K\cap L/K)|$ is a $\pi(HK/K\cap L/K))$-number for any chief factor $L/K$ of $G$, and, if there is a subnormal supplement $T$ of $H$ in $G$ such that $H\cap T\le I\le H$ for some subgroup $I$ satisfying $\Pi$-property in $G$, then $H$ is said $\Pi$-normal in $G$. By these properties of some subgroups, we obtain some new criterions of $p$-nilpotency of finite groups.