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A Cayley graph $\Ga=\Cay(G,S)$ is said to be normal if $G$ is normal in$\Aut\Ga$. The concept of normal Cayley graphs was first proposed by M.Y.Xu in[Discrete Math. 182, 309-319, 1998] and it plays an important role indetermining the full automorphism groups of Cayley graphs. In this paper, weinvestigate the normality problem of the connected arc-transitive pentavalentCayley graphs with soluble vertex stabilizer on finite nonabelian simplegroups. We prove that all such graphs $\Ga$ are either normal or $G=\A_{39}$ or$\A_{79}$. Further, a connected arc-transitive pentavalent Cayley graph on$\A_{79}$ is constructed. To our knowledge, this is the first known example ofpentavalent 3-arc-transitive Cayley graph on finite nonabelian simple groupwhich is non-normal.

Arc-transitive pentavalent Cayley graphs with soluble vertex stabilizer on finite nonabelian simple groups