Study On The Influence Of Quantitative Properties Of Character To The Structures Of Some Almost Simple Groups

In this paper,we first study the relationship between the character degree prime-power graphs and the structure of some almost groups.Atlast,we contin-ue to study whether the number of irreducible characters of symmetric group Sn can be determined by their permutation characters.At the end of this paper,we put forward a conjecture about the relationship between the number of irreducible characters of symmetric group Sn and their permutation characters.This paper is divided into four chapters.In chapter 1,we introduce the research background of this paper.In chapter 2,we give some basic knowledge and some lemmas which will be used in this paper.In chapter 3,we first successfully characterize the two simple groups A8 and L3(4)by using their orders and character degree prime-power graphs.Further-more,we continue to investigate the non-abelian simple groups with 11 ??(G)(?){2,3,5,7,11}.All these above groups having been characterized are non-abelian simple groups.Finally,in the end of this chapter,we successfully characterize the automorphism groups of Mathieu groups by using their orders and their character degree prime-power graphs.In chapter 4,in order to study the relationship between the number of irre-ducible characters of symmetric groups Sn of degree n and their permutation char-acters,we first check these symmetric groups of degree 3 to 10 one by one.Then we put forward a conjecture about the relationship between the number of irreducible characters of symmetric groups Sn of degree n and their permutation characters.