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Cayley graphs of abelian groups which are not normal edge-transitive. (English) Zbl 1124.05040
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 $\vert S\vert \leq 5$ which are edge-transitive but not normal edge-transitive. The proof depends very heavily on the results in [{\it Y.-G. Baik, Y. Feng, H.-S. Sim} and {\it 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 [{\it C. E. Praeger}, Bull. Aust. Math. Soc. 60, 207--220 (1999; Zbl 0939.05047)].

05C25Graphs and abstract algebra
20K01Finite abelian groups
20D99Abstract finite groups