The equivalence of two cyclic objects on pq elements

Abstract

Pálfy showed that two equivalent cyclic objects on n elements are equivalent by a multiplier provided gcd( n, ϕ( n)) = 1 or n = 4. He also showed that there exist equivalent cyclic objects on n elements that are not multiplier equivalent when gcd( n, ϕ( n)) ≠ 1 and n ≠ 4. Huffman et al. (1993) showed that when n = p 2, p an odd prime, two equivalent cyclic objects on n elements are equivalent by elements chosen from a list of at most ϕ( n) permutations. In this paper we show that when n = pq, p and q primes, with gcd( pq, ϕ( pq)) ≠ 1, two equivalent cyclic objects on n elements are equivalent by elements chosen from a list of at most ϕ( n) permutations. In fact, these permutations are products of multipliers and “pieces” of multipliers.