On s-quasinormal and c-normal subgroups of a finite group

Let ℱ be a saturated formation containing the class of supersolvable groups and let G be a finite group. The following theorems are shown: (1) G ∈ ℱ if and only if there is a normal subgroup H such that G/H ∈ ℱ and every maximal subgroup of all Sylow subgroups of H is either c-normal or s-quasinormally embedded in G; (2) G ∈ ℱ if and only if there is a soluble normal subgroup H such that G/H ∈ ℱ and every maximal subgroup of all Sylow subgroups of F(H), the Fitting subgroup of H, is either c-normally or s-quasinormally embedded in G. Supported by the Natural Science Foundation of China and the Natural Science Foundation of Guangxi Autonomous Region (No. 0249001)