On some applications of Fitting like subgroups of finite groups

Abstract

In this paper we study the groups all whose maximal or all Sylow subgroups are K-F-subnormal in their product the with generalizations F * (G) and F(G) of the Fitting subgroup. We prove that a hereditary formation F contains every group all whose Sylow subgroups are K-F-subnormal in their product with F * (G) if and only if F is the class of all σ-nilpotent groups for some partition σ of the set of all primes. We obtain a new characterization of the σ-nilpotent hypercenter, i.e. the F-hypercenter and the largest normal subgroup which K-F-subnormalize all Sylow subgroups coincide if and only if F is the class of all σ-nilpotent groups.