On the partial $\Pi$-property of some subgroups of prime power order 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 subgroups of prime power order satisfying the partial $\Pi$-property on the structure of a finite group.Comment: arXiv admin note: substantial text overlap with arXiv:2304.11451. text overlap with arXiv:1301.6361 by other author