Elementary Abelian p-groups of rank greater than or equal to 4p-2 are not CI-groups

In this paper we prove that an elementary Abelian p-group of rank 4p - 2 is not a CI(2)-group, i.e. there exists a 2-closed transitive permutation group containing two non-conjugate regular elementary Abelian p-subgroups of rank 4p - 2, see Hirasaka and Muzychuk (J. Comb. Theory Ser. A 94(2), 339-362, 2001). It was shown in Hirasaka and Muzychuk (loc cit) and Muzychuk (Discrete Math. 264(1-3), 167-185, 2003) that this is related to the problem of determining whether an elementary Abelian p-group of rank n is a CI-group. As a strengthening of this result we prove that an elementary Abelian p-group E of rank greater or equal to 4p - 2 is not a Cl-group, i.e. there exist two isomorphic Cayley digraphs over E whose corresponding connection sets are not conjugate in Aut E.