Finite groups with a system of generalized central elements

Abstract

Let \(H\) be a normal subgroup of a finite group \(G\). A number of authors have investigated the structure of \(G\) under the assumption that all minimal or maximal subgroups in Sylow subgroups of \(H\) are well-situated in \(G\). A general approach to the results of that kind is proposed in this article. The author has found the conditions for \(p\)-elements of \(H\) under which \(G\)-chief \(p\)-factors of \(H\) are \(\mathfrak{F}\)-central in \(G\).