On the desking completions, theta completions and theta pairs for maximal subgroups*

Abstract

Given a maximal subgroup M of a finite group G,a θ completion of M in G is any subgroup C such that C is not contained in M while MG , the core of M in G, is contained in C and C/MG has no propor normal subgroup of G/MG . By using this concept we can reveal the relationship between the concepts of completions and θ-pairs introduced respectively by Deskins, Mukherjee and Bhattacharya. The concept of maximal θ-completions offers a convenience for us to study the Deskins completions in inductions. We obtain in this paper several results which imply a group to be solvable, supersolvable and nilpotent.