Quotients of polynomial rings and regular t-balanced Cayley maps on abelian groups

TL;DR

This paper clarifies a connection between quotients of polynomial rings and RBCM$_{t}$'s on abelian groups, so as to propose a new approach for classifying R BCM$_t} $'s, and obtains many new results.

Abstract

Given a finite group $Γ$, a regular $t$-balanced Cayley map (RBCM$_{t}$ for short) is a regular Cayley map $\mathcal{CM}(G,Ω,ρ)$ such that $ρ(ω)^{-1}=ρ^{t}(ω)$ for all $ω\inΩ$. In this paper, we clarify a connection between quotients of polynomial rings and RBCM$_{t}$'s on abelian groups, so as to propose a new approach for classifying RBCM$_{t}$'s. We obtain many new results, in particular, a complete classification for RBCM$_{t}$'s on abelian 2-groups.

Theorems & Definitions (21)

• Proposition 1.2
• Remark 1.3
• Remark 1.4
• Theorem 2.1
• proof
• Remark 3.2
• Proposition 3.4
• Example 3.5
• Proposition 3.6
• proof