On Certain Distributive Lattices of Subgroups of Finite Soluble Groups

In this paper, we prove the following result. Let be a saturated formation and Σ a Hall system of a soluble group G. Let X be a w-solid set of maximal subgroups of G such that Σ reduces into each element of X. Consider in G the following three subgroups: the -normalizer D of G associated with Σ; the X-prefrattini subgroup W = W(G,X) of G; and a hypercentrally embedded subgroup T of G. Then the lattice generated by T,D and W is a distributive lattice of pairwise permutable subgroups of G with the cover and avoidance property. This result remains true for the lattice , where V is a subgroup of G whose Sylow subgroups are also Sylow subgroups of hypercentrally embedded subgroups of G such that Σ reduces into V. This work is supported by Proyecto BFM 2001-1667-C03-01