On the $\pi$$\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 $\pi$ denote a set of primes and let ${\mathfrak{G}}_{\pi }$ denote the class of all finite $\pi$-groups. We call the subgroup ${\mathcal{N}}_{{\mathfrak{G}}_{\pi },\mathfrak{F}}(G)$ of $G$ the $\pi \mathfrak{F}$-norm of $G$. A normal subgroup $N$ of $G$ is called $\pi \mathfrak{F}$-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 $\pi$ is $\mathfrak{F}$-central in $G$. Let $Z_{\pi \mathfrak{F}}(G)$ denote the $\pi \mathfrak{F}$-hypercentre of $G$, that is, the product of all $\pi \mathfrak{F}$-hypercentral normal subgroups of $G$. In this paper, we study the properties of the $\mathfrak{H}$-$\mathfrak{F}$-norm, especially of the $\pi \mathfrak{F}$-norm of a finite group $G$. In particular, we investigate the relationship between the ${\pi }^{\prime }\mathfrak{F}$-norm and the $\pi \mathfrak{F}$-hypercentre of $G$.