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Regular Cayley maps for dihedral groups
Journal of Combinatorial Theory Series B ( IF 1.2 ) Pub Date : 2020-12-29 , DOI: 10.1016/j.jctb.2020.12.002
István Kovács , Young Soo Kwon

An orientably-regular map M is a 2-cell embedding of a finite connected graph in a closed orientable surface such that the group AutM of orientation-preserving automorphisms of M acts transitively on the set of arcs. Such a map M is called a Cayley map for the finite group G if AutM contains a subgroup, which is isomorphic to G and acts regularly on the set of vertices. Conder and Tucker (2014) classified the regular Cayley maps for finite cyclic groups, and obtain two two-parameter families M(n,r), one for odd n and one for even n, where n is the order of the regular cyclic group and r is a positive integer satisfying certain arithmetical conditions. In this paper, we classify the regular Cayley maps for dihedral groups in the same fashion. Five two-parameter families Mi(n,r), 1i5, are derived, where 2n is the order of the regular dihedral group and r is an integer satisfying certain arithmetical conditions. For each map Mi(n,r), we determine its valence and covalence, and also describe the structure of the group AutMi(n,r). Unlike the approach of Conder and Tucker, which is entirely algebraic, we follow the traditional combinatorial representation of Cayley maps, and use a combination of permutation group theoretical techniques, the method of quotient Cayley maps, and computations with skew morphisms.



中文翻译:

二面体组的常规Cayley映射

定向规则的地图 中号 是有限连通图在闭合可定向曲面中的2单元嵌入,因此该组 ut中号 的方向保持自同构 中号过渡地作用于该组弧。这样的地图中号如果有限组G被称为Cayley映射ut中号包含一个与G同构的子组,该子组定期作用于一组顶点。Conder和Tucker(2014)对有限循环群的常规Cayley映射进行分类,并获得两个两参数族中号ñ[R,一个代表奇数n,另一个代表偶数n,其中n是规则循环群的阶数,r是满足某些算术条件的正整数。在本文中,我们以相同的方式对二面体组的常规Cayley映射进行分类。五个二参数家庭中号一世ñ[R1个一世5导出,其中2 n是规则二面体组的阶数,r是满足某些算术条件的整数。对于每张地图中号一世ñ[R,我们确定其价和价,并描述该组的结构 ut中号一世ñ[R。与完全代数的Conder和Tucker方法不同,我们遵循Cayley映射的传统组合表示形式,并使用置换组理论技术,商Cayley映射方法以及具有偏态射态的计算的组合。

更新日期:2020-12-29
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