有限群的Sylow-2子群的阶和最高阶元的阶与群的结构

发布时间：2018-05-03 07:32

本文选题：有限群 + 最高阶元　；参考：《西南大学》2017年硕士论文

【摘要】：在有限群的结构研究中,用群的阶,子群的阶,元素的阶得出了若干漂亮的群论性质.如:奠定有限群理论基础的Sylow定理,描述子群阶与群阶性质Lagrange定理,描述p | |G|时,G有p阶元存在的柯西定理,奇数阶可解定理,等等.自上世纪80年代起,以施武杰教授为代表的群论专家开始关注用群的基本数量群的阶和元素阶之集来刻画有限单群,并提出了著名的两个阶刻画全部有限单群的猜想.在该猜想得到解决后,一些学者开始关注减少一些数量作为条件是否仍然可以刻画单群.如:只用群的阶和最高阶元的阶来刻画单群,并得到很多单群的刻画.本文试图用Sylow 2-子群的阶和最高阶元的阶来研究单群,但遗憾的是这种描述已不能成为刻画.如:20阶的Frobenius群的最高阶元的阶是5,2-Sylow子群的阶是4,但恰A5也具有这两个性质.基于此,本文从K3-单群的Sylow 2-子群和最高阶元的阶出发,得出一些群论性质的描述,由于水平有限,只给出了 8个K3-单群中7个单群对应情况的性质,而且也无法得到”充要条件”描述,但作为本文的结论的推论,可以得”群的阶”及”最高阶元的阶”能够刻画这7个单群.
[Abstract]:In the study of the structure of finite groups, some beautiful group theory properties are obtained by using the order of group, the order of subgroup and the order of element. For example, the Sylow theorem which lays the foundation of finite group theory, the Lagrange theorem of subgroup order and group order, the Cauchy theorem where p G has p order element, the odd order solvable theorem, and so on. Since the 1980s, the experts in group theory, represented by Professor Schwujie, have begun to focus on the description of finite simple groups by the order and element order of the basic number groups of groups, and have proposed a famous conjecture of two orders characterizing all finite simple groups. After the conjecture has been solved, some scholars begin to pay attention to whether reducing some quantity as a condition can still characterize a simple group. For example, only the order of a group and the order of the highest order are used to characterize a simple group, and many characterizations of a simple group are obtained. This paper attempts to study a simple group by using the order of Sylow 2-subgroup and the order of the highest order, but unfortunately this description can not be characterized. For example, the order of the highest order of a Frobenius group of order 20 is 4, but the order of a subgroup of 5 ~ 2-Sylow is 4, but A _ 5 also has these two properties. Based on this, from the order of Sylow 2-subgroup and the highest order of K3-simple group, some properties of group theory are obtained. Because of the limited level, we only give the corresponding properties of seven simple groups in eight K3-simple groups. But as a corollary of the conclusion of this paper, we can conclude that the order of a group and the order of the highest order can characterize these seven simple groups.
【学位授予单位】：西南大学
【学位级别】：硕士
【学位授予年份】：2017
【分类号】：O152.1

【参考文献】