Digraphs with small automorphism groups that are Cayley on two nonisomorphic groups
Let Γ = Cay(G, S) be a Cayley digraph on a group G and let A = Aut(Γ). The Cayley index of Γ is |A : G|. It has previously been shown that, if p is a prime, G is a cyclic p-group and A contains a noncyclic regular subgroup, then the Cayley index of Γ is superexponential in p.
We present evidence suggesting that cyclic groups are exceptional in this respect. Specifically, we establish the contrasting result that, if p is an odd prime and G is abelian but not cyclic, and has order a power of p at least p3, then there is a Cayley digraph Γ on G whose Cayley index is just p, and whose automorphism group contains a nonabelian regular subgroup.