The equivalence of two cyclic objects on \(pq\) elements
A cyclic object on \(n\) elements is a combinatorial structure on \(\{0,1,\dots,n-1\}\) which has an \(n\)-cycle in its automorphism group. Examples of classes of cyclic objects are circulant graphs or digraphs or cyclic block designs. For any such class the symmetric group on \(\{0,1,\dots,n-1\}\) induces an equivalence relation. Since most permutations, when applied to a cyclic object, will produce an object that is not cyclic, the equivalence classes are much smaller than in the general case without cyclicity assumption. Depending on the object class, and on the number-theoretic properties of \(n\), it has been shown in many cases that only equivalence by multipliers is possible. Pálfy showed that this is the case for any class of cyclic objects if \(\text{gcd}(n,\varphi(n))=1\). In this paper, the author shows that for \(n=pq\) with \(p\), \(q\) primes in the case \(\text{gcd}(n,\varphi(n))\neq1\) there are still only few possibilities: there is a list of at most \(\varphi(n)\) permutations such that equivalence of cyclic objects on \(n\) elements is possible only by these permutations. They have a piecewise structure of a multiplier.
scientific article; zbMATH DE number 906895
scientific article