Finite CI-groups are soluble

For a finite group G and a subset S of G with 1 is not an element of S and S = S-l, the 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) is an element of S. The group G is called a CI-group if, for all subsets S and T of G\{1}, Cay(G, S) congruent to Cay(G, T) if and only if S-sigma = T for some sigma is an element of Aut(G). In this paper, for each prime p = 1 (mod 4), a symmetric graph Gamma(p) is constructed from PSL(2, p) such that Aut Gamma(p) = Z(2) x PSL(2, p); it is then shown that A(5) is not a CI-group, and that all finite CI-groups are soluble.