On the πF-norm and the H-F-norm of a finite group

Let H be a Fitting class and F a formation. We call a subgroup NH,F(G) of a finite group G the H-F-norm of G if NH,F(G) is the intersection of the normalizers of the products of the F-residuals of all subgroups of G and the H-radical of G. Let π denote a set of primes and let Gπ denote the class of all finite π-groups. We call the subgroup NGπ,F(G) of G the πF-norm of G. A normal subgroup N of G is called πF-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 F-central in G. Let ZπF(G) denote the πF-hypercentre of G, that is, the product of all πF-hypercentral normal subgroups of G. In this paper, we study the properties of the H-F-norm, especially of the πF-norm of a finite group G. In particular, we investigate the relationship between the π′F-norm and the πF-hypercentre of G.