6-valent arc-transitive cayley graphs on abelian groups

Let g be a finite group and s be a subset of g such that 1g ̸∈ s and s −1 = s. the cayley graph σ = cay(g, s) on g with respect to s is the graph with the vertex set g such that, for §, † ∈ g, the pair (§, †) is an arc in cay(g, s) if and only if †§−1 ∈ s. the graph σ is said to be arc-transitive if its full automorphism group aut(σ) is transitive on its arc set. in this paper we give a classification for arc-transitive cayley graphs with valency six on finite abelian groups which are non-normal. moreover, we classify all normal cayley graphs on non-cyclic abelian groups with valency 6.