边传递映射的乘积分解
Parallel-product decomposition of edge-transitive maps
摘要：
S. E. 1994年，威尔逊本文的主要问题是，对于给定的自反映射$M$，能否将该映入两个自反映射的平行积。当且仅当映射的单值（或自同构）群至少有两个极小正规子群。所有的自反映射到100个边缘，这是不是平行产品可分解，计算和提出。为此目的，我们对所有退化和轻度退化自反映射进行了分类。本文考虑了根映射的三种不同的同构，给出了根映射态射的一个类似于群的第一同构定理的特征。引入了一个映射的单值商，它具有所有自同构都投影的性质。建立了非可定向曲面上的边传递映射理论。在Breda d'Azevedo的方式减少正则性的概念应用于边传递映射。在此基础上，将平行积可分解的概念推广到边传递映射，得到了自同构群的极小正规子群的一个刻画。此外，使用Petrie triality和平行积分解，一个新的组织的边传递映射，提供了一个基础，为未来的人口普查。
英文摘要：
The parallel product of two rooted maps was introduced by S. E. Wilson in 1994. The main question of this paper is whether for a given reflexible map $M$ one can decompose the map into a parallel product of two reflexible maps. This can be achieved if and only if the monodromy (or the automorphism) group of the map has at least two minimal normal subgroups. All reflexible maps up to 100 edges, which are not parallel-product decomposable, are calculated and presented. For this purpose, all degenerate and slightly-degenerate reflexible maps are classified. Three different quotients of rooted maps are considered in the paper and a characterizaton of morphisms of rooted maps similar to the first isomorphism theorem for groups is presented. The monodromy quotient of a map is introduced, having the property that all the automorphisms project. A theory of edge-transitive maps on non-orientable surfaces is developed. A concept of reduced regularty in the manner of Breda d'Azevedo is applied on edge-transitive maps. Using that, the concept of parallel-product decomposability is extended to edge-transitive maps, where a characterization in terms of minimal normal subgroups of the automorphism group is obtained. Additionally, using Petrie triality and the parallel-product decomposition, a new organization of edge-transitive maps is presented, providing a basis for future censuses.
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