Conjugacy Class Sizes And Generalized Normal Subgroups Of Finite Groups

The relationships between the arithmetical conditions on conjugacy classes of a finite group and the structure of the finite group itself have been extensively studied by many authors in the past two decades.In particular,many results have been obtained by many group theorists in the literature and the research of finite groups has been pushed forward by these fruitful results contained in this topic.Among the research of arithmetical conditions,it is very active to use both of the sizes and the number of conjugacy classes.In this thesis,we concentrate on the investigation of their mutual relationships.The main objectives of this thesis are the following:Ⅰ.Investigating the structure of a finite group G in which the conjugacy classes of some elements of G having a particular arithmetical property.Ⅱ.Investigating the structure of a finite group G when the number of the sizes of eonjugaey classes of some elements in the group is a fixed number.There are two way for one to study objectiveⅠat present.One is to reduce the number of the conjugaey classes of some elements,the other is to study the conjugacy classes of some elements of G having a more general arithmetical property.As the continuation of these two kinds of research way,in this thesis, some known corresponding results in objectiveⅠare generalized.In Chapter 3, we study the cases that the sizes of the conjugacy classes of some p′-elements with prime-power order are square-free-number,cube-free-number or p-number, respectively,where p is a prime.In Chapter 6,we characterize the structure of G when its sizes of the conjugacy classes of all elements are cube-free-number.For the objectiveⅡ,many authors have already studied the structure of a finite group in which the number of the sizes of its conjugacy classes is a fixed number.Based on these results,one further investigated the structure of the p-complement by assuming that the number of the sizes of the conjugacy classes of some p-regular elements is a fixed number.Recently,some authors considered the finite groups in which the number of the sizes of its conjugacy classes of elemnts with prime-power order is a fixed number.From these results,we would naturally ask whether we can reduce the number of corresponding conjugacy classes when we consider objectiveⅡ?Since the basic elements in a finite group are the elements with prime-power order,and the elements of biprimary and triprimary orders are only slightly complex elements compared to elements of prime-power order in a finite group, we hope to investigate the structure of finite groups by using the number of the sizes of the conjugacy classes of some elements with prime-power,biprimary and triprimary orders.This is the aim of this thesis.In fact,in Chapter 4,the structure of finite groups is investigated if the number of the sizes of the conjugacy classes of some elements of prime-power, biprimary and triprimary orders is 2 or 4,some known result of Ito is generalized. By taking the above results in consideration,in Chapter 5,we characterize in detail the structure of G having three special conjugacy classes sizes,a known result of Ito on three conjugacy classes sizes is partially generalized.Thus,in this thesis,we have thoroughly studied the conjugacy classes of a finite group and we have given some descriptions of the finite groups by considering the properties of its some conjugacy classes.It is noted that if we do not consider all the conjugacy classes of a given group G,then we call this idea the localization of a finite group.In the last Chapter of this thesis,we shall apply the idea of localization of a finite group to study the relationship between the structure of a finite group and 3-permutability of some subgroups in a local sub-group, and give some sufficient conditions for a finite group to be a p-nilpotent or supersolvable group.