Finite groups with permutable complete Wielandt sets of subgroups

Abstract Let 𝒲 be a set of nilpotent Hall subgroups of a finite group G. We say that 𝒲 is a Wielandt set if for each prime p dividing |G| there is exactly one subgroup W ∈ 𝒲 such that p divides |W|. In this paper, we study the influence of Wielandt sets on the structure of G. In particular, we give conditions under which a normal subgroup of a finite group is hypercyclically embedded.