FINITE GROUPS WITH GIVEN NEARLY s-EMBEDDED SUBGROUPS

Let G be a finite group and H a subgroup of G. We say that H is s-permutable in G if HP = PH for all Sylow subgroups P of G; H is s-semipermutable in G if HP = PH for all Sylow subgroups P of G with (|P|, |H|) = 1. Let H(s)G be the subgroup of H generated by all those subgroups of G which are s-permutable in G and H-sG the intersection of all such s-permutable subgroups of G contain H. We say that H is nearly s-embedded in G if G has an s-permutable subgroup T such that H-sG = HT and H boolean AND T <= H(ss)G, where H(ss)G is an s-semipermutable subgroup of G contained in H. In this paper, we study the structure of a finite group G under the assumption that some subgroups of prime power order are nearly s-embedded in G. A series of known results are improved and extended.