On the isomorphism problem for finite Cayley graphs of bounded valency

For a subset S of a group G such that 1 is not an element of 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) epsilon S. Each sigma epsilon Aut(G) induces an isomorphism from Cay(G, S) to the Cayley graph Cay(G, S-sigma). 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) congruent to Cay(G, T) there is an element sigma epsilon Aut(G) such that S-sigma = T. It is shown that if G is an m-CI-group for some m greater than or equal to 4, then G = U x 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. (C) 1999 Academic Press.