A note on $p$-supersolubility of finite groups

Abstract

Let $H$ be a subgroup of a finite group $G$. We say that $H$ satisfies $\mathcal{L}$-$\Pi$-property in $G$ if $|G/K:N_{G/K}(HK/K)|$ is a $\pi (HK/K)$-number for all maximal $G$-invariant subgroup $K$ of $H^G$. In this paper, we give a characterization of finite $p$-supersoluble groups under assumption that some subgroups of prime power order satisfy $\mathcal{L}$-$\Pi$-property.