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Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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Regular Cayley maps for cyclic groups
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by Marston D.E. Conder and Thomas W. Tucker PDF
Trans. Amer. Math. Soc. 366 (2014), 3585-3609 Request permission

Abstract:

An orientably-regular map $M$ is a 2-cell embedding of a connected graph in a closed, orientable surface, with the property that the group $\mathrm {Aut}^\textrm {o}M$ of all orientation-preserving automorphisms acts transitively on the arcs of $M$. If $\mathrm {Aut}^\textrm {o}M$ contains a subgroup $A$ that acts regularly on the vertex set, then $M$ is called a regular Cayley map for $A$. In this paper, we answer a question of recent interest by providing a complete classification of the regular Cayley maps for the cyclic group $C_n$, for every possible order $n$. This is the first such classification for any infinite family of groups. The approach used is entirely algebraic and does not involve skew morphisms (but leads to a classification of all skew morphisms which have an orbit that is closed under inverses and generates the group). A key step is the use of a generalisation by Conder and Isaacs (2004) of Ito’s theorem on group factorisations, to help determine the isomorphism type of $\mathrm {Aut}^\textrm {o}M$. This group is shown to be a cyclic extension of a cyclic or dihedral group, dependent on $n$ and a single parameter $r$, which is a unit modulo $n$ that satisfies technical number-theoretic conditions. For each $n$, we enumerate all such $r$, and then in terms of $r$, we find the valence and covalence of the map, and determine whether or not the map is reflexible, and whether it has a representation as a balanced, anti-balanced or $t$-balanced regular Cayley map.
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Additional Information
  • Marston D.E. Conder
  • Affiliation: Department of Mathematics, University of Auckland, Private Bag 92019, Auckland 1142, New Zealand
  • MR Author ID: 50940
  • ORCID: 0000-0002-0256-6978
  • Thomas W. Tucker
  • Affiliation: Department of Mathematics, Colgate University, Hamilton, New York 13346
  • MR Author ID: 175090
  • ORCID: 0000-0002-7868-6925
  • Received by editor(s): March 2, 2011
  • Received by editor(s) in revised form: August 9, 2012
  • Published electronically: March 3, 2014
  • © Copyright 2014 American Mathematical Society
    The copyright for this article reverts to public domain 28 years after publication.
  • Journal: Trans. Amer. Math. Soc. 366 (2014), 3585-3609
  • MSC (2010): Primary 05E18; Secondary 05C10, 20B25, 57M15
  • DOI: https://doi.org/10.1090/S0002-9947-2014-05933-3
  • MathSciNet review: 3192608