A generalization of Hall's theorem on hypercenter

Abstract

Let σ be a partition of the set of all primes and F be a hereditary formation. We proved that the F-hypercenter and the intersection of weak K-F-subnormalizers of all Sylow subgroups coincide if and only if F is the class of all σ-nilpotent groups. As corollary we obtained P. Hall's theorem about the intersection of normalizers of Sylow subgroups.