On p-nilpotency of finite groups?

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.