On isomorphisms of abelian Cayley objects of certain orders

AbstractLet m be a positive integer such that gcd(m,ϕ(m))=1 (ϕ is Euler's 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

Classification of Cyclic Steiner Quadruple Systems

The problem of classifying cyclic Steiner quadruple systems (CSQSs) is considered. A computational approach shows that the number of isomorphism classes of such designs with orders 26 and 28 is 52,170 and 1,028,387, respectively. It is further shown that CSQSs of order 2p, where p is a prime, are isomorphic iff they are multiplier equivalent. Moreover, no CSQSs of order less than or equal to 38 are isomorphic but not multiplier equivalent