【作者】 赵燕；

【导师】 邵长国；

【作者基本信息】 济南大学 ， 应用数学， 2021， 硕士

【摘要】 群是数学中广泛存在的一个重要概念,它是一种具有运算的非空代数系.群在抽象代数中的地位是极其重要的,同时也是最基本的,环、模和域等代数结构都可以看作是在群的基础上添加新的运算和公理而形成的.在有限群的研究中,如何确定有限群的结构是一个重要的研究内容,而群的数量刻画理论就是研究群结构的重要工具.而单群作为有限群的一个重要内容,引起了许多群论工作者的研究兴趣.对于群的数量刻画,人们总是希望可以用最少的数量刻画出群最多的性质.而在有限群固有的众多的数量关系中,人们对共轭类的某些数量性质,如:共轭类个数,共轭类长,共轭类长度的个数等有着特殊的兴趣.本篇论文的内容如下:首先,介绍有限群论数量刻画的研究背景及运用共轭类刻画群和对单群刻画的研究现状,对群论的研究有一个大概的认识.其次,我们从与类长相等的元素个数这一数量性质出发,先得到了群G是非可解,再通过分析与群的主因子同构的单群,进而对单群₅A进行了刻画.之后我们通过群的共轭类长这一数量性质对单群PSL₂（7）7（8）进行了一个刻画,先是得到了群的阶的素因子的集合,再通过证明得到群中没有某些固定阶的元,最后通过群是非可解的得到结论.再次,已知群的结构与它的实共轭类长的算数性质之间存在一定的关系,所以我们对实共轭类长集合为{1,2}的有限群G的结构进行了证明.在证明过程中我们得到了只有两个实共轭类长的群是可解的,再通过群的Sylow 2-子群与群的正规2-补这两个子群的相互作用得到了群的结构性质.考虑完实共轭类长后,我们又对共轭类长只有一个合数m的有限群G进行了研究,在定理的证明过程中我们通过对元素中心化子的幂零性的分析得到了该群的结构性质.最后是对本篇论文研究内容的总结,提出了本文的创新点.更多还原

【Abstract】 The group is an important concept widely existing in mathematics.It is a kind of non empty algebra system with operation.The position of group in abstract algebra is extremely important and basic,such as rings,modules and fields can be regarded as the result of adding new operations and axioms to the group.In the study of finite groups,how to determine the structure of finite groups is an important research content,and the quantitative characterization theory of groups is an important tool to study the structure of groups.As an important content of finite groups,simple groups have aroused the research interest of many group theorists,People always hope that the most properties of a group can be characterized by the least number.Among the numerous quantitative relations inherent in finite groups,people have special interest in some quantitative properties of conjugate classes,such as the number of conjugate classes,the length of conjugate classes,the number of length of conjugate classes,etc.The contents of this paper are as follows:First of all,we introduce the background of the development of the quantitative characterization of finite group theory and the research status of using conjugate classes to characterize groups and simple groups,so as to have a general understanding of the research of group theory.Secondly,starting from the quantitative property of the number of elements equal to the class length,we first obtain that the group G is non-solvable,and then characterize the simple groupA₅ by analyzing the simple group isomorphic to the main factor of the group.Then we characterize the simple groupPSL₂（7）7（8）by the quantitative property of the conjugacy class length of the group,and first obtain the set of prime factors of the order of the group,Finally,we get the conclusion that the group is not solvable.Thirdly,we know that there is a certain relationship between the structure of a group and the arithmetic property of its real conjugacy class length,so we prove the structure of a finite group G whose real conjugate class length set is{1,2}.In the process of proof,we get that only two groups with real conjugacy class length are solvable.Then we obtain the structural properties of the group by the interaction between the Sylow 2-subgroup and the normal2-complement of the group.After considering the length of the real conjugate class,we study the finite group G whose conjugate class has only one composite number m.In the process of proving the theorem,we obtain the structural properties of the group by analyzing the nilpotency of the element centralizer.Finally,the summary of the research content of this paper and the innovation of this paper.更多还原