On c -Normal Maximal and Minimal Subgroups of Sylow Subgroups of Finite Groups. II

Abstract

A subgroup H of G is said to be c-normal in G if there exists a normal subgroup N of G such that HN = G and H ∩ N ≤ H G = Core(H). We extend the study on the structure of a finite group under the assumption that all maximal or minimal subgroups of the Sylow subgroups of the generalized Fitting subgroup of some normal subgroup of G are c-normal in G. The main theorems we proved in this paper are: Theorem Let ℱ be a saturated formation containing 𝒰. Suppose that G is a group with a normal subgroup H such that G/H ∈ ℱ. If all maximal subgroups of any Sylow subgroup of F*(H) are c-normal in G, then G ∈ ℱ. Theorem Let ℱ be a saturated formation containing 𝒰. Suppose that G is a group with a normal subgroup H such that G/H ∈ ℱ. If all minimal subgroups and all cyclic subgroups of F*(H) are c-normal in G, then G ∈ ℱ.