Finite groups with some subgroups satisfying the partial $\Pi$-property

Abstract

Let $H$ be a subgroup of a finite group $G$. We say that $H$ satisfies the partial $\Pi$-property in $G$ if there exists a chief series ${\Gamma }_G:1=G_0<G_1<\cdot \cdot \cdot <G_n=G$ of $G$ such that for every $G$-chief factor $G_i/G_{i-1}$ $(1\le i\le n)$ of ${\Gamma }_G$, $|G/G_{i-1}:N_{G/G_{i-1}}(H{G}_{i-1}/G_{i-1}\cap G_i/G_{i-1})|$ is a $\pi (H{G}_{i-1}/G_{i-1}\cap G_i/G_{i-1})$-number. In this paper, we investigate how some subgroups satisfying the partial $\Pi$-property influence the structure of finite groups.