A Cayley graph Γ=Cay(G

S)is said to be normal if Gis normal in Aut(Γ). The concept of normal Cayley graphs was first proposed by XU Ming Yao and it plays an important role in determining the full automorphism groups of Cayley graphs. The Cayley graphs on finite nonabelian simple groups are received most attention in the literature. However

examples of connected arctransitive nonnormal Cayley graphs of small valency on nonabelian simple groups are very rare. An example of a nonnormal 2-transitive pentavalent Cayley graph on the alternating group A

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is constructed and it is showed  that the full automorphism group of this graph is isomorphic to the alternating group A

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.