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

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.