On the Intersection of the Normalizers of the -Residuals of Subgroups of a Finite Group

In this paper we give criteria for a finite group to belong to a formation. As applications, recent theorems of Li, Shen, Shi and Qian are generalized. Let G be a finite group, a formation and p a prime. Let be the intersection of the normalizers of the -residuals of all subgroups of G, and let be the intersection of the normalizers of for all subgroups H of G. We then define and , . Let and denote the terminal member of the ascending series of and respectively. In this paper we prove that under certain hypotheses, the the -residual is nilpotent (respectively,p-nilpotent) if and only if (respectively, ). Further more, if the formation is either the class of all nilpotent groups or the class of all abelian groups, then is p-nilpotent if and only if and only if every cyclic subgroup of G order p and 4 (if p = 2) is contained in .