Cayley graphs of abelian groups which are not normal edge-transitive
Let \(G\) be a group and let \(S\subseteq G\) with \(1_G\not\in S\). The Cayley graph \(\text{Cay}(G,S)\) is the graph with vertex set \(G\) and (directed) edge set \(E=\{(x,y)\mid yx^{-1}\in S\}\). If \(S^{-1}=S\), then \((x,y)\) is an edge of \(\text{Cay}(G,S)\) if and only if \((y,x)\) is an edge of \(\text{Cay}(G,S)\); in this case, \(\text{Cay}(G,S)\) is said to be undirected and the pair \(\{(x,y), (y,x)\}\) is called an unordered edge. The group \(G\) acts transitively on the vertices of \(\Gamma=\text{Cay}(G,S)\) via right translations \(\rho_g: x\to xg\), and \(G_R=\{\rho_g\mid g\in G\}\leq\text{Aut}(\Gamma)\) can be identified with \(G\). Let \(\text{Aut}(G,S)=\{\sigma\in\text{Aut}(G)\mid\sigma(S)=S\}\). Then \(N_{\text{Aut}(\Gamma)}(G)=G\rtimes\text{Aut}(G,S)\). A Cayley graph \(\Gamma=\text{Cay}(G,S)\) is called edge-transitive if \(\text{Aut}(\Gamma)\) acts transitively on the edges (or the unordered edges) of \(\Gamma\). A Cayley graph \(\Gamma=\text{Cay}(G,S)\) is normal edge-transitive if \(N_{\text{Aut}(\Gamma)}(G)\) acts transitively on the edges (or unordered edges) of \(\Gamma\). In this paper the authors classify those pairs \((G,S)\) where \(G\) is an abelian group and \(S\) is a symmetric generating set with \(| S| \leq 5\) which are edge-transitive but not normal edge-transitive. The proof depends very heavily on the results in [\textit{Y.-G. Baik, Y. Feng, H.-S. Sim} and \textit{M. Xu}, Algebra Colloq. 5, No.~3, 297--304 (1998; Zbl 0904.05037)]. The authors give a list of 13 cases, they list the group \(G\), the set \(S\) and the graph. Many of the essential properties of these are found in [\textit{C. E. Praeger}, Bull. Aust. Math. Soc. 60, 207--220 (1999; Zbl 0939.05047)].
scientific article; zbMATH DE number 5156823