On solubility and supersolubility of some classes of finite groups

Let G be a finite group and H a subgroup of G. Then H is said to be S-permutable in G if HP = PH for all Sylow subgroups P of G. Let HsG be the subgroup of H generated by all those subgroups of H which are S-permutable in G. Then we say that H is S-embedded in G if G has a normal subgroup T and an S-permutable subgroup C such that T ∩ H HsG and HT = C. Our main result is the following Theorem A. A group G is supersoluble if and only if for every non-cyclic Sylow subgroup P of the generalized Fitting subgroup F*(G) of G, at least one of the following holds: (1) Every maximal subgroup of P is S-embedded in G. (2) Every cyclic subgroup H of P with prime order or order 4 (if P is a non-abelian 2-group and H Z∞(G)) is S-embedded in G.