On finite groups with the Cayley isomorphism property

Let G be a finite group, and let Cay(G, S) be a Cayley digraph of G. If, for all T subset of G, Cay(G, S) congruent to Cay(G, T) implies S-alpha = T for some alpha is an element of Aut(G), then Cay(G, S) is called a CI-graph of G. For a group G, if all Cayley digraphs of valency m are CI-graphs, then G is said to have the m-DCI property; if all Cayley graphs of valency m are CI-graphs, then G is said to have the m-CI property. It is shown that even/finite group of order greater than 2 has a nontrivial CI-graph, and all finite groups with the m-CI property and with the m-DCI property are characterized for small values of m. A general investigation is made of the structure of Sylow subgroups of finite groups with the m-DCI property and with the m-CI property for large values of m. (C) 1998 John WiIey & Sons, Inc.MathematicsSCI(E)7ARTICLE121-312