Character Degrees And The Structures Of Finite Groups

It is a classic and important subject to study the structures of finite groups by character theory of finite groups,such as Frobenius theorem and p^aq^b theorem.Many scholars have studied the relationship about the character degrees and the structures of finite groups,and given many important results.In this paper,we consider the structures of finite groups by the arithmetical properties of cd(G) and the character degree graph.It consists of the following four chapters:In chapter 1,we introduce some symbols and basic concepts that we usually use in the paper.In chapter 2,we study finite groups whose all irreducible character degrees are Hall-numbers.And we have given the structures of these groups.In chapter 3,we study finite groups with the character degree prime graphs containing no triangles.In chapter 4,we study a kind of subgraphs of prime graphs,i.e.the graphs afforded by |cd(G)|-1 character degrees.We define the graphâ–³(G-m),whose vertices are the elements ofρ(G-m)={p|p|a,m≠a∈cd(G)},i.e.those primes q that divide some element of cd(G)\{m},where m∈cd(G) is a positive integer.We draw an edge between two different vertices q,r∈△(G-m) if and only ifqr divides a for some a∈cd(G)\{m}. The number of connected components ofâ–³(G-m) will be denoted by n(â–³(G-m)).We know that if G is an abelian group,or cd(G)={1,a} and m=a,then cd(G)\{m}=φ, so let n(â–³(G-m))=0.We have proved that n(â–³(G-m))≤2 to each m∈cd(G),if G is a solvable group or G≌A_n,where n≥7(A₅≌L₂(4)≌L₂(5),A₆≌L₂(9)),or G is one of the sporadic simple groups.And n(â–³(G-m))≤3 to each m∈cd(G),if G is a finite simple group of Lie type.