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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 ${1}_{G}\notin S$. The Cayley graph $\text{Cay}\left(G,S\right)$ is the graph with vertex set $G$ and (directed) edge set $E=\left\{\left(x,y\right)\mid y{x}^{-1}\in S\right\}$. If ${S}^{-1}=S$, then $\left(x,y\right)$ is an edge of $\text{Cay}\left(G,S\right)$ if and only if $\left(y,x\right)$ is an edge of $\text{Cay}\left(G,S\right)$; in this case, $\text{Cay}\left(G,S\right)$ is said to be undirected and the pair $\left\{\left(x,y\right),\left(y,x\right)\right\}$ is called an unordered edge. The group $G$ acts transitively on the vertices of ${\Gamma }=\text{Cay}\left(G,S\right)$ via right translations ${\rho }_{g}:x\to xg$, and ${G}_{R}=\left\{{\rho }_{g}\mid g\in G\right\}\le \text{Aut}\left({\Gamma }\right)$ can be identified with $G$. Let $\text{Aut}\left(G,S\right)=\left\{\sigma \in \text{Aut}\left(G\right)\mid \sigma \left(S\right)=S\right\}$. Then ${N}_{\text{Aut}\left({\Gamma }\right)}\left(G\right)=G⋊\text{Aut}\left(G,S\right)$. A Cayley graph ${\Gamma }=\text{Cay}\left(G,S\right)$ is called edge-transitive if $\text{Aut}\left({\Gamma }\right)$ acts transitively on the edges (or the unordered edges) of ${\Gamma }$. A Cayley graph ${\Gamma }=\text{Cay}\left(G,S\right)$ is normal edge-transitive if ${N}_{\text{Aut}\left({\Gamma }\right)}\left(G\right)$ acts transitively on the edges (or unordered edges) of ${\Gamma }$. In this paper the authors classify those pairs $\left(G,S\right)$ where $G$ is an abelian group and $S$ is a symmetric generating set with $|S|\le 5$ which are edge-transitive but not normal edge-transitive. The proof depends very heavily on the results in [Y.-G. Baik, Y. Feng, H.-S. Sim and 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 [C. E. Praeger, Bull. Aust. Math. Soc. 60, 207–220 (1999; Zbl 0939.05047)].
##### MSC:
 05C25 Graphs and abstract algebra 20K01 Finite abelian groups 20D99 Abstract finite groups
##### Keywords:
Cayley graph; edge-transitive; abelian group