On the $π$$\mathfrak{F}$-norm and the $\mathfrak{H}$-$\mathfrak{F}$-norm of a finite group

Abstract

Let $\mathfrak{H}$ be a Fitting class and $\mathfrak{F}$ a formation. We call a subgroup $\mathcal{N}_{\mathfrak{H},\mathfrak{F}}(G)$ of a finite group $G$ the $\mathfrak{H}$-$\mathfrak{F}$-norm of $G$ if $\mathcal{N}_{\mathfrak{H},\mathfrak{F}}(G)$ is the intersection of the normalizers of the products of the $\mathfrak{F}$-residuals of all subgroups of $G$ and the $\mathfrak{H}$-radical of $G$. Let $π$ denote a set of primes and let $\mathfrak{G}_π$ denote the class of all finite $π$-groups. We call the subgroup $\mathcal{N}_{\mathfrak{G}_π,\mathfrak{F}}(G)$ of $G$ the $π\mathfrak{F}$-norm of $G$. A normal subgroup $N$ of $G$ is called $π\mathfrak{F}$-hypercentral in $G$ if either $N=1$ or $N>1$ and every $G$-chief factor below $N$ of order divisible by at least one prime in $π$ is $\mathfrak{F}$-central in $G$. Let $Z_{π\mathfrak{F}}(G)$ denote the $π\mathfrak{F}$-hypercentre of $G$, that is, the product of all $π\mathfrak{F}$-hypercentral normal subgroups of $G$. In this paper, we study the properties of the $\mathfrak{H}$-$\mathfrak{F}$-norm, especially of the $π\mathfrak{F}$-norm of a finite group $G$. In particular, we investigate the relationship between the $π'\mathfrak{F}$-norm and the $π\mathfrak{F}$-hypercentre of $G$.