Normal and non-normal Cayley digraphs on cyclic and dihedral groups

TL;DR

The paper investigates when cyclic and dihedral groups admit NNND- and NNN-Cayley digraphs/graphs, focusing on the existence and structure of non-normal regular subgroups in automorphism groups. It develops a concise, self-contained approach using the holomorph and fixed-point subgroups to classify these phenomena for $C_n$ and $D_{2n}$. It proves that no cyclic group can be NNND, while giving a complete dihedral classification: $D_{2n}$ is NNND/NNN iff $n\ge 6$ is even with $n\neq 8$. The results deepen understanding of when a group supports a normal Cayley digraph with a non-normal regular copy of the group, with implications for Cayley isomorphism properties in this context.

Abstract

A Cayley digraph on a group $G$ is called NNN if the Cayley digraph is normal and its automorphism group contains a non-normal regular subgroup isomorphic to $G$. A group is called NNND-group or NNN-group if there is an NNN Cayley digraph or graph on the group, respectively. In this paper, it is shown that there is no cyclic NNND-group, and hence no cyclic NNN-group. Furthermore, a dihedral group of order $2n$ is an NNND-group or an NNN-group if and only if $n\ge 6$ is even and $n\not=8$.

Figures (2)

• Figure 1: The induced subgraph $[\Gamma_2[1]]$
• Figure 2: The induced subgraph $[Ma^i\cup Ma^{i+1}\cup Ma^{-i}b\cup Ma^{-i-1}b]$.

Theorems & Definitions (10)

• Theorem 1.1
• Theorem 1.2
• Proposition 2.1
• Proposition 2.2
• Proposition 2.3
• Proposition 2.4
• Lemma 3.1
• Lemma 3.2
• Lemma 4.1
• Lemma 4.2