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Regular balanced Cayley maps on ${\rm PSL}(2,p)$

Haimiao Chen

Abstract

A {\it regular balanced Cayley map} (RBCM for short) on a finite group $Γ$ is an embedding of a Cayley graph on $Γ$ into a surface, with some special symmetric property. People have classified RBCM's for cyclic, dihedral, generalized quaternion, dicyclic, and semi-dihedral groups. In this paper we classify RBCM's on the group ${\rm PSL}(2,p)$ for each prime number $p>3$.

Regular balanced Cayley maps on ${\rm PSL}(2,p)$

Abstract

A {\it regular balanced Cayley map} (RBCM for short) on a finite group is an embedding of a Cayley graph on into a surface, with some special symmetric property. People have classified RBCM's for cyclic, dihedral, generalized quaternion, dicyclic, and semi-dihedral groups. In this paper we classify RBCM's on the group for each prime number .

Paper Structure

This paper contains 8 sections, 11 theorems, 27 equations.

Table of Contents

  1. Introduction
  2. Preliminary
  3. Type I regular balanced Cayley maps
  4. I-RBCM's on ${\rm PSL}(2,5)=A_{5}$
  5. I-RBCM's on ${\rm PSL}(2,p)$ for $p>5$
  6. Type II regular balanced Cayley maps
  7. II-RBCM's on ${\rm PSL}(2,5)=A_{5}$
  8. II-RBCM's on ${\rm PSL}(2,p)$ for $p>5$

Key Result

Proposition 1.1

(a) A Cayley map $\mathcal{CM}(\Gamma,\Omega,\rho)$ is a RBCM if and only if $\rho$ extends to an isomorphism of $\Gamma$. (b) For a RBCM $\mathcal{CM}(\Gamma,\Omega=\{\omega_{i}\colon 1\leq i\leq m\},\rho)$ with $\rho(\omega_{i})=\omega_{i+1}$, all of the elements of $\Omega$ have the same order, a A RBCM in the case (I) or (II) is said to be of type I or II, and denoted by I-RBCM or II-RBCM, res

Theorems & Definitions (15)

  • Proposition 1.1
  • Proposition 2.1
  • Proposition 2.2
  • Remark 2.3
  • Proposition 2.4
  • Theorem 3.1
  • Lemma 3.2
  • proof
  • Theorem 3.3
  • Theorem 4.1
  • ...and 5 more