On subgroups of hypercentral type of finite groups

The main purpose of this paper is to analyze the influence on the structure of a finite group of some subgroups lying in the hypercenter. More precisely, we prove the following: Let \(\mathfrak{F}\) be a Baer-local formation. Given a group G and a normal subgroup E of G, let \(Z_\mathfrak{F} (G)\) contain a p-subgroup A of E which is maximal being abelian and of exponent dividing p ^k , where k is some natural number, k ≠ 1 if p = 2 and the Sylow 2-subgroups of E are non-abelian. Then E/O _p′ (E) ≤ \(Z_\mathfrak{F} \) (G/O _p′ (E)) (Theorem 1). Some well-known results turn out to be consequences of this theorem.