On the deskins completions and theta completions for maximal subgroups

Let G be a finite group and M a maximal subgroup of G. A θ-completion of M in G is any subgroup C such that C is not contained in M while M G , the core of M in G, is contained in C and has no proper normal subgroup of . The concept of θ -completion offers a convenience for us to study the Deskins completions. By using this concept and in a quite different way from what was used, we obtain some new results about the maximal completions and θ-completions which imply G to be solvable and supersolvable.