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Abstract. For a group G, and a subset S of G such that 1G  S,
let Γ = Cay(G, S) be the corresponding Cayley
graph. Then Γ is said to be normal edge transitive, if NAut(Γ)(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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