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Vietnam Journal of Mathematics 33:3 (2005) 309-318  Cayley Graphs of Abelian Groups Which Are Not Normal Edge-Transitive Mehdi Alaeiyan, Hamid Tavallaee, and Ali A. Talebi Abstract.  For a group $G$, and a subset $S$ of $G$ such that $1_{G} \not \in S$, let $\Gamma = Cay(G,S)$ be the corresponding Cayley graph. Then $\Gamma$ is said to be normal edge transitive, if $N_{Aut(\Gamma)}(G)$ is transitive on edges. In this paper we determine all connected, undirected edge-transitive Cayley graphs of finite abelian groups with valency at most five, which are not normal edge transitive. This is a partial answer to a question of Praeger.

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