【作者】 游兴中；

【导师】 施武杰；

【作者基本信息】 苏州大学 ， 应用数学， 2005， 博士

【摘要】 本文主要研究有限群中共轭类和元素的阶的算术条件对群结构的影响。 第一章主要介绍和本文工作相关的文献背景及本文要解决的问题。 第二章主要研究有限群的中心外的同阶元的共轭类个数对群结构的影响。我们首先研究了奇数阶的同阶元均共轭的有理群的结构，其次刻画了中心外的同阶元必共轭的有限群的结构。我们得到的结果是W．Feit，G．M．Seits及张继平所得的一个结论的推广，此外我们还给出了Syskin问题的一个简洁的新证明。 第三章主要研究有限群的正规子群外的共轭类的个数对群结构的影响。设G为有限群，N为G的一个正规子群。我们研究了当G中至多有3个G-共轭类不在N中时群G的结构。 第四章主要研究有限群的对偶图的算术条件对群结构的影响。对偶于E．A．Bertram等人定义的共轭类图，我们自然地定义了有限群G的对偶图Γ（G）：它以G的非中心的元的共轭类作为顶点，任何两个不同的顶点D（=x^G）和C（=y^G）之间有一条边相连当且仅当o（x）与o（y）有非平凡的公因子。若Γ（G）中不含有由n个顶点构成的其中任意两点连通的完全子图，我们称G有性质P_n。我们研究了Γ（G）的连通分支数和直径，分别刻画了具有性质P₃和P₄的有限群的结构。 第五章主要研究有限群中元素的阶的互素性对群结构的影响。设G是有限群，π_e（G）表示G的所有元素的阶的集合。若G的任意n个互不相同的元素的阶是互素的，即对任意互不相同的n个元素a₁，a₂，…，a_n∈π_e（G），满足（a₁，a₂，…，a_n）=1，我们说G有性质Φ_n。我们给出了满足性质Φ₃的有限群的分类。更多还原

【Abstract】 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 π_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.更多还原