Finite groups with some subgroups of prime power order 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 $G$-chief series $ \varGamma_{G}: 1 =G_{0} < G_{1} < \cdot\cdot\cdot < G_{n}= G $ of $ G $ such that $ | G / G_{i-1} : N_{G/G_{i-1}} (HG_{i-1}/G_{i-1}\cap G_{i}/G_{i-1})| $ is a $ \pi (HG_{i-1}/G_{i-1}\cap G_{i}/G_{i-1}) $-number for every $ G $-chief factor $ G_{i}/G_{i-1} $ of $ \varGamma_{G} $, $1\leq i\leq n$. In this paper, we investigate the structure of a finite group $ G $ under the assumption that some subgroups of prime power order satisfy the partial $ \Pi $-property.