On sublattices of the subgroup lattice defined by formation Fitting sets.

Throughout this paper, G always denotes a group and L (G) is the lattice of all subgroups of G. If K ⊴ H ≤ G , then H / K is called a section of G ; such a section is called normal if K , H ⊴ G. We call any set Σ of normal sections of G a stratification of G provided: (i) Σ is G-closed , that is, H / K ∈ Σ whenever H / K ≃ G T / L ∈ Σ , and (ii) L / K , H / L ∈ Σ for each triple K < L < H , where H / K ∈ Σ and L ⊴ G. Now let Σ be any stratification of G. Then we write L Σ (G) to denote the set of all subgroups A of G such that A G / A G ∈ Σ. We say that a stratification Σ of G is a formation Fitting set of G provided: (i) H / (K ∩ N) ∈ Σ for every two sections H / K , H / N ∈ Σ , and (ii) H V / K ∈ Σ for every two sections H / K , V / K ∈ Σ. We prove that the set L Σ (G) forms a sublattice of L (G) for every formation Fitting set Σ of G and we also discuss some applications of sublattices of this kind. [ABSTRACT FROM AUTHOR]