两类正则地图

发布时间：2018-03-05 08:28

本文选题：正则凯莱地图　切入点：中心对称　出处：《烟台大学》2016年硕士论文　论文类型：学位论文

【摘要】：本文主要研究了正则地图理论中的两个问题：一是哪些群上存在中心对称正则凯莱地图,二是哪些群可以作为莫比乌斯正则地图的自同构群.对于第一个问题,我们证明了两类具有循环极大子群的有限p-群上不存在中心对称正则凯莱地图.该结果和M. Conder, Y. Wang等人的结论一起对具有循环极大子群的有限p-群上的中心对称正则凯莱地图的存在与否给出了回答.本文使用了作“商”这一代数学处理手法,使得大部分的计算都在商群中进行,比这些文献中处理这类问题常用的证明方法更为简练易懂,并且我们的证明方法还可以在其它的有限p-群中进行推广对于第二个问题,我们证明了对称群S6不能作为莫比乌斯正则地图的自同构群；而在同构意义下,S7可以作为2个莫比乌斯正则地图的自同构群；S8可以作为3个莫比乌斯正则地图的自同构群.事实上,决定一个群G是否可以作为莫比乌斯正则地图的自同构群,我们需要对G的子群进行分析.而本章则是利用子群的结构和关系,将问题转化为对G中元素的阶进行分析.注意到我们考虑的群是对称群,因此这种转化的好处是显然的,我们利用对元素阶的分析决定了对称群S6,S7,S8是否可以作为莫比乌斯正则地图的自同构群,这为我们进一步研究此类地图提供一定的方法.
[Abstract]:In this paper, we mainly study two problems in the theory of regular maps: first, which groups have centrosymmetric regular Calais maps, and second, which groups can be used as automorphism groups of Mobius regular maps. We prove that there is no centrosymmetric canonical Calais map on two classes of finite p- groups with cyclic maximal subgroups. This result, together with the results of M. Conder, Y. Wang et al., shows the central symmetry of finite p- groups with cyclic maximal subgroups. The existence or not of canonical Calais map is answered. In this paper, the mathematical processing technique of "quotient" is used. So that most of the calculations are done in quotient groups, which is much simpler and easier to understand than the proof methods used to deal with this kind of problem in these documents, and our proof method can also be generalized in other finite p- groups for the second problem. We prove that symmetric group S6 is not an automorphism group of Mobius regular map. In the sense of isomorphism, S7 can be used as an automorphism group of two Mobius regular maps and S8 can be used as an automorphism group of three Mobius regular maps. To determine whether a group G can be regarded as an automorphism group of a Mobius regular map, we need to analyze the subgroups of G. in this chapter, we use the structure and relations of subgroups. The problem is transformed into an analysis of the order of elements in G. notice that the group we are considering is a symmetric group, so the benefits of this transformation are obvious. We use the analysis of element order to determine whether the symmetric group S6 S7 S8 can be regarded as the automorphism group of Mobius regular map, which provides a certain method for us to further study this kind of map.
【学位授予单位】：烟台大学
【学位级别】：硕士
【学位授予年份】：2016
【分类号】：O152.1

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