On the partial $\Pi$-property of second minimal or second maximal subgroups of Sylow subgroups of finite groups

Abstract

Let $ H $ be a subgroup of a finite group $ G $. We say that $ H $ satisfies the partial $ \Pi $-property in $ G $ if if there exists a chief series $ \varGamma_{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\leq i\leq n) $ of $ \varGamma_{G} $, $ | 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. In this paper, we study the influence of some second minimal or second maximal subgroups of a Sylow subgroup satisfying the partial $ \Pi $-property on the structure of a finite group.