The normality of Cayley graphs of finite abelian groups with valency 5 (Q2721960)

A Cayley graph \(X= \text{Cay}(G, S)\) is said to be normal if the right regular representation of \(G\) is a normal subgroup of the automorphism group of \(X\). The concept of normality of Cayley graphs is important for the study of arc-transitive graphs. If \(G\) is a cyclic group or a group of order \(2p\) (\(p\) prime), then all normal or non-normal Cayley (di)graphs are known. The main result of the paper is a classification of all non-normal connected Cayley graphs of finite abelian groups with valency 5. In a previous paper the authors characterized all non-normal connected Cayley graphs of finite abelian groups with valency at most 4; see \textit{Y.-G. Baik}, \textit{Y. Feng}, \textit{H.-S. Sim}, and \textit{M. Xu} [Algebra Colloq. 5, No. 3, 297-304 (1998; Zbl 0904.05037)].