Sizes And Numbers Of Conjugacy Classes And The Structure Of Finite Groups

The conjugacy class sizes and conjugacy class numbers of a finite group have close relationship with the structure of a finite group, and many group workers have participated in this area, and lots of important results have been obtained. In recent years, many workers try to determine the structure of a finite group G by using the numbers of G-conjugacy classes contained in its proper normal subgroups. One part of this thesis is to investigate the structure of a finite group G when the numbers of G-conjugacy classes contained in its all non-trivial normal subgroups are three consecutive integers. In fact, in Chapter3, we investigate the structure of a finite group G when the set of numbers of G-conjugacy classes contained in its all proper normal subgroups is{1,2,3,4}, and we give a complete classification of finite non-perfect groups which satisfy this condition. Furthermore, we also investigate the structure of a finite group G when the set of numbers of G-conjugacy classes contained in its all proper normal subgroups is{1, m, m+1, m+2}, and we obtain the structure of a finite non-perfect group in this case (where m is an arbitrary integer larger than2).Another part of this thesis is to investigate the structure of a finite group G by using the conjugacy class sizes of part of its elements. Suppose that N is a normal subgroup of G. We mainly investigate the structure of N by using the G-conjugacy class sizes of some elements in N. In fact, in Chapter4, by using the properties of conjugacy class graph of a finite group, we investigate the relationship between the G-conjugacy class sizes of p’-elements in a p-solvable normal subgroup N and the structure of p-complements of N. In Chapter5, we investigate the influence of the G-conjugacy class sizes of primary and biprimary7r’-elements in a normal subgroup N of a7r-separable group G on the structure and properties of7r-complements of N.