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 described all formations F for which the F-hypercenter and the intersection of weak K-Fsubnormalizers of all Sylow subgroups coincide in every group. In particular the formation of all σ-nilpotent groups has this property. With the help of our results we solve a particular case of L.A. Shemetkov’s problem about the intersection of F-maximal subgroups and the F-hypercenter. As corollaries we obtained P. Hall’s and R. Baer’s classical results about the hypercenter. We proved that the non-σ-nilpotent graph of a group is connected and its diameter is at most 3.