Arithmetical Conditions And The Structures Of Finite Groups

This thesis mainly considers how the arithmetical conditions of conjugacy classes and 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 investigate how the number of conjugacy classes of elements outside the center with the same order of a finite group influences its structure. We first study the structures of the rational groups in which elements with the same odd order are conjugate and then characterize the finite groups in which elements outside the center with the same order are conjugate, our result is a generalization of a result obtained by W. Feit, G. M. Seits and J. P. Zhang respectively, we also give an alternative proof of the Syskin problem.In Chapter 3, we investigate how the number of conjugacy classes outside a normal subgroup of a finite group influences its structure. Let G be a finite group and N a normal subgroup of G. We study the structures of the group G when there are at most three conjugacy classes of G outside N.In Chapter 4, we exhibit how the arithmetical conditions of the dual graph of a finite group influence its structures. Dualizing to a graph related to conjugacy classes of a finite group G defined by E. A. Bertram etc, we define a dual graph Γ(G) of G: its vertices set is the set of non-central conjugacy classes of G, any two distinct vertices D(= x^G) and C(= y^G) are connected by an edge if and only if o(x) and o(y), the orders of x and y, have a nontrivial common divisor. We say G satisfies the property P_n, if Γ(G) contains no a complete subgraph consisting of n vertices in which any two ones are connected. We study the number of connected components and the diameter of Γ(G), classify the finite groups with the property P₃ and P₄ respectively.In Chapter 5, we study how the co-primeness of element orders of a finite group influences its structure. Let Ï€_e(G) be the set of all element orders of a finite group G. We say G satisfies the property Φ_n, if any n distinct element orders are setwise coprime, that is, for any n distinct elements a₁,a₂,…… ,a_n ∈π_e(G), (a₁,a₂,……,a_n) = 1. We classify the finite groups with the property Φ₃.