Summary: In [Eur. J. Comb. 8, 35-43 (1987; Zbl 0614.05049)] P. P. Pálfy answers the question: Under what conditions on \(n\) is it true that two equivalent objects in any class of cyclic combinatorial objects on \(n\) elements implies the objects are equivalent, using one of the \(\varphi(n)\) multipliers \(i\to ai\bmod n\), with \(\text{gcd}(a,n)=1\). Pálfy proved that this is true precisely when \(n=4\) or \(\text{gcd}(n,\varphi(n))=1\). For any odd prime \(p\), we prove that two equivalent objects in any class of cyclic combinatorial objects on \(n=p^ 2\) elements are equivalent using a permutation from a list of no more then \(\varphi(n)=p(p-1)\) permutations. We introduce permutations called generalized multipliers, and we show that two permutation equivalent cyclic codes of length \(p^ 2\) are equivalent by a generalized multiplier times a multiplier. We also develop properties of generalized multipliers and generalized affine maps when \(n=p^ m\), show that they map cyclic codes to cyclic codes, and show that certain of these maps are in the automorphism group of a cyclic code.