Regular Cayley maps on dihedral groups with the smallest kernel

TL;DR

This paper classifies regular Cayley maps on dihedral groups based on the kernel of their power function, identifying minimal kernel cases and describing specific families and sporadic examples.

Contribution

It proves the kernel of the power function is a dihedral subgroup and classifies maps with minimal kernel, revealing new infinite families and sporadic cases.

Findings

Kernel of the power function is a dihedral subgroup

Kernel size is at least 4 for most cases

Identifies two infinite families and four sporadic maps

Abstract

Let $\mathcal{M}=CM(D_n,X,p)$ be a regular Cayley map on the dihedral group $D_n$ of order $2n, n \ge 2,$ and let $\pi$ be the power function associated with $\mathcal{M}$. In this paper it is shown that the kernel Ker$(\pi)$ of the power function $\pi$ is a dihedral subgroup of $D_n$ and if $n \ne 3,$ then the kernel Ker$(\pi)$ is of order at least $4$. Moreover, all $\mathcal{M}$ are classified for which Ker$(\pi)$ is of order $4$. In particular, besides $4$ sporadic maps on $4,4,8$ and $12$ vertices respectively, two infinite families of non-$t$-balanced Cayley maps on $D_n$ are obtained.