On the isomorphism problem for finite Cayley graphs of bounded valency

Abstract

For a subset S of a group G such that 1 ∉ S and S = S-1, the associated Cayley graph Cay(G, S) is the graph with vertex set G such that {x, y} is an edge if and only if yx-1 ∈ S. Each σ ∈ Aut(G) induces an isomorphism from Cay(G, S) to the Cayley graph Cay(G, Sσ). For a positive integer m, the group G is called an m-CI-group if, for all Cayley subsets S of size at most m, whenever Cay(G, S) ≅ Cay(G, T) there is an element σ ∈ Aut(G) such that Sσ = T. It is shown that if G is an m-CI-group for some m ≥ 4, then G = U × V, where (|U|, |V|) = 1, U is abelian, and V belongs 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. © 1999 Academic Press.