On $\pi \mathfrak{F}$-supplemented subgroups of a finite group

Abstract

Let $\mathfrak{F}$ be a class of groups and $G$ a finite group. A chief factor $H/K$ of $G$ is called $\mathfrak{F}$-central in $G$ provided $(H/K)⋊(G/C_G(H/K))\in \mathfrak{F}$. A normal subgroup $N$ of $G$ is said to be $\pi \mathfrak{F}$-hypercentral in $G$ if every chief factor of $G$ below $N$ of order divisible by at least one prime in $\pi$ is $\mathfrak{F}$-central in $G$. The $\pi \mathfrak{F}$-hypercentre of $G$ is the product of all the normal $\pi \mathfrak{F}$-hypercentral subgroups of $G$. In this paper, we study the structure of finite groups by using the notion of $\pi \mathfrak{F}$-hypercentre. New characterizations of some classes of finite groups are obtained.