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Let $\mathfrak{H}$ be a Fitting class and $\mathfrak{F}$ a formation. We calla 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 thenormalizers 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 andlet $\mathfrak{G}_\pi$ denote the class of all finite $\pi$-groups. We call thesubgroup $\mathcal{N}_{\mathfrak{G}_\pi,\mathfrak{F}}(G)$ 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 studythe properties of the $\mathfrak{H}$-$\mathfrak{F}$-norm, especially of the$\pi\mathfrak{F}$-norm of a finite group $G$. In particular, we investigate therelationship between the $\pi'\mathfrak{F}$-norm and the$\pi\mathfrak{F}$-hypercentre of $G$.

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