On automorphisms of Cayley-digraphs of abelian groups Original Research Article

The automorphisms of the Cayley-digraph Γ are considered, where Γ is defined according to some abelian group G. Using Fourier transformation defined over G, we derive an equation relating automorphisms of Γ and characters of G. Then our result is used to estimate the number of simple eigenvalues that Γ can have, when it admits a non-regular automorphism group. As another application of our method, a short proof is also shown for a classical theorem of W. Burnside on transitive permutation groups of prime degree.