On ℱ-Quasinormal Primary Subgroups of Finite Groups

Let G be a finite group and ℱ a formation. A subgroup H is called ℱ-quasinormal in G if there exists a quasinormal subgroup T of G such that HT is quasinormal in G and (H ∩ T)H G /H G is contained in the ℱ-hypercenter of G/H G . In this article, we study the structure of finite groups by using ℱ-quasinormal subgroups and prove that: Let ℱ be a saturated formation containing 𝒰 and G be a group with a normal subgroup H such that G/H ∈ ℱ. If every maximal subgroup of every noncyclic Sylow subgroup of F*(H) having no supersolvable supplement in G is 𝒰-quasinormal in G, then G ∈ ℱ.