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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 SG with 1 G S. The Cayley graph Cay(G,S) is the graph with vertex set G and (directed) edge set E={(x,y)yx -1 S}. If S -1 =S, then (x,y) is an edge of Cay(G,S) if and only if (y,x) is an edge of Cay(G,S); in this case, 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 Γ=Cay(G,S) via right translations ρ g :xxg, and G R ={ρ g gG}Aut(Γ) can be identified with G. Let Aut(G,S)={σAut(G)σ(S)=S}. Then N Aut(Γ) (G)=GAut(G,S). A Cayley graph Γ=Cay(G,S) is called edge-transitive if Aut(Γ) acts transitively on the edges (or the unordered edges) of Γ. A Cayley graph Γ=Cay(G,S) is normal edge-transitive if N Aut(Γ) (G) acts transitively on the edges (or unordered edges) of Γ. 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|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:
05C25Graphs and abstract algebra
20K01Finite abelian groups
20D99Abstract finite groups