On the Isomorphism Problem for Finite Cayley Graphs of Bounded Valency

For a subsetSof a groupGsuch that 1 ∉ SandS = S−1, the associated Cayley graph Cay(G, S) is the graph with vertex setGsuch that {x, y} is an edge if and only ifyx−1 ∈ S. Each σ ∈ Aut (G) induces an isomorphism from Cay(G, S) to the Cayley graph Cay(G, Sσ). For a positive integerm, the groupGis called anm-CI-group if, for all Cayley subsetsSof size at mostm, whenever Cay(G, S) ≅ Cay(G, T) there is an element σ ∈ Aut(G) such thatSσ = T. It is shown that ifGis anm-CI-group for somem ≥ 4, thenG = U × V, where (|U|, |V|) = 1,Uis abelian, andVbelongs to an explicitly determined list of groups. Moreover, Sylow subgroups of such groups satisfy some very restrictive conditions. This classification yields, as corollaries, improvements of results of Babai and Frankl. We note that our classification relies on the finite simple group classification.