映射与超映射的手征群
Chirality Groups of Maps and Hypermaps
摘要：
虽然手征现象出现在许多研究的地图和超地图，没有详细的研究手征似乎已经进行。映射和超映射的手征性不仅是一个二元不变量，而且可以用两个新的不变量来量化--手征性群和手征性指数，后者是手征性群的大小。本文的主要目的是研究映射和超映射的手征群。当手征群与单值群重合时，出现最极端的手征类型。这种超映射称为全手征映射。它们的例子是通过考虑适当的“非对称”对的生成元的一些非阿贝尔单群。我们还证明了每个有限阿贝尔群都是某些超映射的手征群，而许多非阿贝尔群，包括对称群和二面体群，都不能作为手征群出现。
英文摘要：
Although the phenomenon of chirality appears in many investigations of maps and hypermaps no detailed study of chirality seems to have been carried out. Chirality of maps and hypermaps is not merely a binary invariant but can be quantified by two new invariants -- the chirality group and the chirality index, the latter being the size of the chirality group. A detailed investigation of the chirality groups of maps and hypermaps will be the main objective of this paper. The most extreme type of chirality arises when the chirality group coincides with the monodromy group. Such hypermaps are called totally chiral. Examples of them are constructed by considering appropriate ``asymmetric'' pairs of generators for some non-abelian simple groups. We also show that every finite abelian group is the chirality group of some hypermap, whereas many non-abelian groups, including symmetric and dihedral groups, cannot arise as chirality groups.
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