An Elementary Abelian Group of Rank 4 Is a CI-Group

Let Cay(H, S), S/H be an arbitrary Cayley graph over a finite group H. We shall say that two Cayley graphs Cay(H, S) and Cay(H, T) are Cayley isomorphic if there exists an automorphism . # Aut(H ) such that S=T. It is a trivial observation that two Cayley isomorphic Cayley graphs are isomorphic as graphs. The converse is not true: two Cayley graphs over the same group may be isomorphic as graphs but not Cayley isomorphic. There are many examples of this phenomenon, see, for example, [2, 11]. A subset S H is called a CI-subset if for each T H the graphs Cay(H, S), Cay(H, T) are isomorphic if and only if the sets T and S are conjugate by an element of Aut(H ). A group H is called a CI-group if each subset of H is a CI-subset. doi:10.1006 jcta.2000.3140, available online at http: www.idealibrary.com on