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Regular $t$-balanced Cayley maps on split metacyclic $2$-groups

Haimiao Chen, Jingrui Zhang

Abstract

A regular $t$-balanced Cayley map on a group $Γ$ is an embedding of a Cayley graph on $Γ$ into a surface with certain special symmetric properties. We completely classify regular $t$-balanced Cayley maps for a class of split metacyclic $2$-groups.

Regular $t$-balanced Cayley maps on split metacyclic $2$-groups

Abstract

A regular -balanced Cayley map on a group is an embedding of a Cayley graph on into a surface with certain special symmetric properties. We completely classify regular -balanced Cayley maps for a class of split metacyclic -groups.

Paper Structure

This paper contains 7 sections, 13 theorems, 70 equations.

Table of Contents

  1. Introduction
  2. Preliminary on metacyclic groups
  3. Classifying regular $t$-balanced Cayley maps for a class of split metacyclic 2-groups
  4. Constraints
  5. Normalization
  6. Expressing necessary and sufficient conditions in terms of congruence equations
  7. The result

Key Result

Proposition 1.1

(a) A Cayley map $\mathcal{CM}(\Gamma,\Omega,\rho)$ is a RBCM$_{1}$ if and only if $\rho$ can be extended to an automorphism of $\Gamma$. (b) Suppose $t>1$. A Cayley map $\mathcal{CM}(\Gamma,\Omega,\rho)$ is a RBCM$_{t}$ if and only if $\rho$ can be extended to a skew-morphism of $\Gamma$, $\pi(\ome

Theorems & Definitions (25)

  • Proposition 1.1
  • Remark 1.2
  • Lemma 1.3
  • proof
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • ...and 15 more