On finite groups with some primary subgroups satisfying partial $S$-$\Pi$-property

Abstract

A $p$-subgroup $H$ of a finite group $G$ is said to satisfy partial $S$-$\Pi$-property in $G$ if $G$ has a chief series ${\Gamma }_G:1=G_0<G_1<\cdots <G_n=G$ such that for every $G$-chief factor $G_i/G_{i-1}$ $(1⩽i⩽n)$ of ${\Gamma }_G$, either $(H\cap G_i)G_{i-1}/G_{i-1}$ is a Sylow $p$-subgroup of $G_i/G_{i-1}$ or $|G/G_{i-1}:N_{G/G_{i-1}}((H\cap G_i)G_{i-1}/G_{i-1})|$ is a $p$-number. In this paper, we mainly investigate the structure of finite groups with some primary subgroups satisfying partial $S$-$\Pi$-property.