Some Problems Of Supersolubility Of Finite Groups And The Product Of Two Normal Supersoluble Subgroups

Supersoluble groups are an important kind of groups. This doctoral thesis main-ly study some problems about the supersolubility of finite groups. The content can be divided into there parts:1, we study the influence of the action of a Frobenius group on the group G on the supersolubility and p-supersolubility of G, and partially solve an open problem.2, we obtain new characteristics of the hypercyclicity of the normal sub-groups of G and the supersolubility of G through studying the partial C4P*-subgroups of some groups.3, we classify the structures of the minimal non-supersoluble groups which are the product of two normal supersoluble subgroups. This makes us to solve a difficult problem.This thesis is divided into 5 chapters.In Chapter 1, we briefly recall the historical origins of the groups theory, the de-velopments of the theory of finite groups. In addition, we introduce the main results of this thesis.In Chapter 2, we introduce the definitions of some basic notions and the basic knowledge of the theory of finite groups.In Chapter 3, we study the case of a finite group G with a Frobenius group FH as an automorphism group such that CG(F)= 1. At first, we establish the connection be-tween the properties of G and CG{H) by proving that G is p-closed (resp. p-nilpotent) if CG(H) is p-closed (resp. p-nilpotent). E. I.Khukhro, N. Y. Makarenko and P. Shumy-atsky [45, Lemma 2.6] proved that:if p ∈π(G), then G has the unique FH-invariant Sylow p-subgroup. In this chapter, we improve this result:G does not only has those unique FH-invariant Sylow subgroups, but also those FH-invariant Sylow subgroups form a Sylow system of G. At the end of this chapter, we use established results to prove that:if CG(H) is p-supersoluble and GG(H) is p-nilpotent, then G is p-supersoluble. Therefore, we can obtain a characteristic of the supersolubility of G. As a consequence, this partially solve Khukhro’s problem.In Chapter 4, based on the CAP-subgroups, partial CMP-subgroups and CAP*-subgroups we further study the influence of partial CAP*-subgroups on the structure of finite groups. At first, we study some basic properties of partial CMP*-subgroups. Afterward, we study the influence of partial CAP*-subgroups on the supersolubility and p-nilpotency of finite groups, based on assumption that the maximal subgroups of the Sylow subgroups of certain groups are partial CAP*-subgroups. At the end, we construct an example to show that if every minimal and 4 order cyclic subgroups of G is a partial CAP*-subgroup of G, then G is not necessary be supersoluble. Moreover, we will give some applications of our results of this chapter.In Chapter 5, we study the classification of the minimal non-supersoluble groups which are the product of two normal supersoluble subgroups. It lead us to solve an open difficult problem. We used this result to find a necessary and sufficient condition of the product of two normal supersoluble subgroups to be supersoluble. Moreover, we used also find some sufficient conditions of the product of two normal supersoluble subgroups to be supersoluble.