On the deskins completions, theta completions and theta pairs for maximal subgroups II*

Abstract

This paper is a continuation of the paper “On the Deskins completions, theta completions and theta pairs for maximal subgroups ”.In the former paper, Zhao Yaoqing introduced the concept of θ-completions associated to a maximal subgroup of a finite group. The concept offers a convenience for us to study the completions introduced by Deskins and gives us a way to reveal the relationship between the concepts of completions and θ-pairs, the latter concept is introduced by Mukherjee and Bhattacharya. The present paper is devoted to discussing the π-solvability, π-supersolvability and π-nilpotency of a finite group by using the θ-completions. Moreover, a new proof on the Deskins conjecture concerning the supersolvability is included.