(sic)τ-Embedded and (sic)τΦ-Embedded Subgroups of Finite Groups

Let (sic) be a nonempty formation of groups, tau a subgroup functor, and H a p-subgroup of a finite group G. Suppose also that G = G/H-G and H = H/H-G. We say that H is (sic)(tau)-embedded ((sic)(tau Phi)-embedded) in G if, for some quasinormal subgroup T of G and some tau-subgroup S of G contained in H, the subgroup HT is S-quasinormal in G and H boolean AND T <= SZ((sic)Phi)(G) (resp., H boolean AND T <= SZ((sic)Phi)(G)). Using the notions of (sic)(tau)-embedded and (sic)(tau)-Phi-embedded subgroups, we give some characterizations of the structure of finite groups. A number of earlier concepts and related results are further developed and unified.