Conjugacy Class Sizes And The Normal Structure Of Finite Groups

The relationships between the arithmetical condition on conjugacy classes of a finite group and the structure of a 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 theoretists in the literature.The research of finite groups has been pushed forwarded by these fruitful results contained in this topic.It is noteworthy in the arithmetical condition on conjugacy classes,both their sizes and the numbers of their sizes are closely related and linked.In this thesis,we concentrate on the investigation of their mutual relationships.The main objectives of this thesis are the following:Ⅰ.To investigate the structure of a subgroup of a finite group G when the number of the sizes of the G-conjugacy class of some elements in the subgroup is a fixed number;Furthermore,one would ask whether we can possibly reduce the number of its conjugacy classes having a fixed size?Ⅱ.To investigate the structure of a finite group G in which the conjugacy class of some elements of G having a particular arithmetical property.For the objectiveⅠ,many authors have already studied the structure of a finite group in which the number of the size of its conjugacy class is a fixed number.Based on these results,people further investigated the structure of the p-complement by assuming that the number of the length of the conjugacy class of some p-regular elements is a fixed number.Recently,there are some authors who have considered the finite groups in which the number of the sizes of its conjugacy class of elemnts having a 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 objectiveⅠ? If N is a normal subgroup of a finite group G,then it is known that N is a union of some conjugacy classes of G.Obviously,the number of the G-conjugacy classes in N is not more than the number of the conjugacy classes contained in G.In order to reduce the number of the conjugacy classes,we consider the G-conjugacy classes contained in N,In other words,we study the structure of N by assuming that the number of the sizes of the conjugacy class of some elements in N is a fixed number.This is the aim of this thesis.In Chapter 3,we study the structure of the normal subgroup of G by assuming that the number of its G-conjugacy class sizes is 2.We concentrate on normal subgroups having exactly two G-conjugacy class sizes,in particular,a known result of Ito is generalized.Based on the results in Chapter 3,we are able to further reduce the number of conjugacy classes and consider the case that the number of the sizes of conjugacy class of elements with order of prime power in N is equal to 2 in Chapter 4.By taking the above results in consideration, we also study the relationships between the number of the sizes of the p-regular G-conjugacy classes and their corresponding p-complements of N.Thus the main question arisen inⅠis answered.In Chapter 6,some known corresponding results in objectiveⅡare generalized. In particular,we study the cases that the lengths of the conjugacy class of some prime power orderπ′-elements areπ′-number,π-number orτ-number, respectively,whereτ=π∪{q,r},πis a set of some primes,q and r are distinct primes with q,r(?)π.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 andπ-quasi-normality of some subgroups in a local subgroup,and to generalize some related results in the saturated group formations containing some supersolvable group classes.