The Influence Of The Normal Indice Of Subgroups And The Subgroups Generated By Conjugate Subgroup Pair On The Structure Of Finite Groups

The aim of this Ph.D.dissertation is to study the influence of the normal indice of subgroups and the subgroups generated by conjugate subgroup pair on the structure of finite groups.This paper is organized as the following five chapters:In Chapter 1, we introduce some notation, basic concepts, and some results that we often use in the paper.Chapter 2 is devoted to investigating the structure of finite groups by using the normal index and c-section of the maximal subgroups. Let M be a maximal subgroup of a finite group G, the order of a chief factor H/K where H is a minimal supplement to M in G is called the normal index of M, and M (?) H/K is called a c-section of M. Using the concepts of the normal index and the c-section, we obtain the characterizations for a finite group to be solvable, p-supersolvable and supersolvable.In Chapter 3, we extend the normal index from maximal subgroups to proper sub-groups. We give a quantitative version of all results obtained by using c-normal subgroups and obtain some new characterizations of solvable, supersolvable and nilpotent groups by the normal indices of proper subgroups.In Chapter 4, we investigate the properties of G from the properties of the subgroups (H, H9) and the indices of the subgroups H in (H, H9) for all g (?) G, where H is either a 2-maximal subgroup of G or a Sylow subgroup of G.The purpose of Chapter 5 is to investigate the structure of G by using the concept of theθ*-pair. We get some new characterizations of a finite group being solvable or supersolvable.