On the Cayley Isomorphism Problem for Cayley Objects of Nilpotent Groups of Some Orders

We give a necessary condition to reduce the Cayley isomorphism problem for Cayley objects of a nilpotent or abelian group G whose order satisfies certain arithmetic properties to the Cayley isomorphism problem of Cayley objects of the Sylow subgroups of G in the case of nilpotent groups, and in the case of abelian groups to certain natural subgroups. As an application of this result, we show that Zq×Zp×Zm is a CI-group with respect to digraphs, where q and p are primes with p2 < q and m is a square-free integer satisfying certain arithmetic conditions (but there are no other restrictions on q and p).