On isomorphisms of connected Cayley graphs

Let G be a finite group and Cay(G,S) the Cayley graph of G with respect to S. A subset S is called a CI-subset if, for any 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). Ln this paper, we investigate the finite groups G in which every subset S with size at most m and (S) = G is a CI-subset where m is a positive integer. As a corollary, we classify symmetric graphs of order p(3) and of valency 2p where p is a prime.