The Arithmetical Conditions Of Conjugacy Classes And The Structures Of Finite Groups

The concept of prime graph arose during the investigation of certain cohomo-logical questions associated with integral representations of finite groups. In 1975, K. W. Gruenberg and K. W. Roggenkamp proved that the prime graph of a finite group G is not connected if and only if the augmentation ideal of G is decomposable as a module. After that, some problems that how some properties of the graph influence the structure of the finite group, regarded as hot ones, studied by many group theorists, the work of the graph of finite simple group and its structure, s-tudied by J. S. William, M. Suzuki, A. S. Kondrat’ev, N. Iiyora and H.Yamaki etc., is the most important. Later, group theorists found that some other graphs, such as the graph of degrees of characters and the graph of conjugacy classes of finite group ect., can defined, and they can characterize the structure of the finite group by some properties of the these graphs. Thus defining some graphs and studying how some properties of these graphs influence the structure of the finite group becomes a important topic in finite group theory.This thesis mainly considers how the properties of prime graph and graph of element orders of a finite group influence its structure respectively.In Chapter 1, we mainly introduce the works related to this thesis and problems. that will be solved in this thesis.In Chapter 2, we give some necessary preliminaries, including some basic con-ceptions,lemmas and their proofs.In Chapter 3, we investigate how the properties of prime graph of a finite group influence its structure. Let G be a finite group, denote by π(G’) the set of prime divisors of the order of G and 7re(G) the set of element orders of G. The prime graph Γ1(G) of G is defined as follows:the set of vertices is π(G) and two vertices p, q are joined by an edge if and only if pq ∈ Tre(G). We investigate the structure of finite solvable group when the number of connected components of its prime graph and give a upper bound of its diameter of the prime graph, also we character the finite group when its prime graph contains no a loop.In Chapter 4, we investigate how the properties of graph of element orders of a finite group influence its structure. Let G be a finite group, the graph of element orders Γ2(G) of G is defined as follows:the set of vertices is πe(G) and two vertices α, b are joined by an edge if and only if (a, b)= ρ for some prime p. We say G has property Pn if Γ2(G) has no subgraph consist of n vertices. We investigate the structures of finite solvable groups and nonabelian finite simple groups with property P4 respectively.