Group The Nature Of The Character And Group Structure

It is an important subject to study the structures of finite groups by character theory of finite groups. The character of a finite group plays an important role in determining the structure of a group, and the study on this topic has been being done since the character theory was set up. In this paper, we study the relations of the large character degrees and the structures of a finite group firstly, and then we study the relations of the distribution of zeros in the character table of a finite group and the structures of finite groups. It consists of 4 chapters and the main results are in chapters 3 and 4.In chapter 3, we study the relations of the large character degrees and the struc-tures of finite groups, and try to the new characterization of some finite nonabelian simple groups by their orders and their'large'character degrees. Except 7 simple groups, every K3-simple groups, every Mathieu simple group and every Janko sim-ple group can be characterized by its order and its largest character degree of the group. And all K3-simple groups, Mathieu simple groups and Janko simple groups can be characterized by their orders, their largest and the second largest character degrees. For 7 non-abelian simple group that cannot be characterized by their orders and largest degrees of characters, it is classified finite groups having the same order and largest degree of character with A6 or M22.In chapter 4, we study the relations of the distribution of zeros in the character table and the structures of finite groups, and get the classification of solvable groups having at most p+1 zeros in each column of the character table (where p is the minimal prime divisor of|G|) firstly. And then we study the relations of the lengths of vanishing-congjugacy classes and the structures of finite groups. For the group with length of every vanishing-congjugacy class a power of a prime for, we come to that these groups are solvable. Particularly, it is proved that a group is super-solvable if the length of every vanishing-congjugacy class is a prime number.