On isomorphisms of abelian Cayley objects of certain orders Original Research Article

Let m be a positive integer such that gcd(m,ϕ(m))=1 (ϕ is Eulerʹs phi function) with m=p1⋯pr the prime power decomposition of m. Let n=p1a1⋯prar. We provide a sufficient condition to reduce the Cayley isomorphism problem for Cayley objects of an abelian group of order n to the prime power case. In the case of Cayley k-ary relational structures (which include digraphs) of abelian groups, this sufficient condition reduces the Cayley isomorphism problem of k-ary relational structures of abelian groups to the prime power case for Cayley k-ary relational structures of abelian groups. As corollaries, we solve the Cayley isomorphism problem for Cayley graphs of Zn (for the specific values of n as above) and prove several abelian groups (for specific choices of the ai) of order n are CI-groups with respect to digraphs.