Finite groups with given s-embedded and n-embedded subgroups

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 and HsG the intersection of all such s-permutable subgroups of G which contain H. We say that: (1) H is s-embedded in G if G has an s-permutable subgroup T such that T∩H⩽HsG and HT=HsG; (2) H is n-embedded in G if G has a normal subgroup T such that T∩H⩽HsG and HT=HG.Our main results here are the following theorems.Theorem A – A group G is supersoluble if and only if every maximal subgroup of every non-cyclic Sylow subgroup of the generalized Fitting subgroup F∗(G) of G is n-embedded in G.Theorem B – 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, every cyclic subgroup H of P with prime order and with order 4 (if P is a non-abelian 2-group and H⊈Z∞(G)) is n-embedded in G.Theorem F – A group G is supersoluble if and only if every 2-maximal subgroup E of G with non-primary index |G:E|, both has a cyclic supplement in EsG and is s-embedded in G.