(Prior to publication, this paper should be cited as arXiv:math/0103046.)

We study the map  c→g(c)  on  Z / pⁿZ,  where  g(X)  is a polynomial with integer coefficients and  p  is an odd prime. If  g  permutes  Z / pⁿZ,  we show how the cycle structure of this permutation is determined by the cycle structure of the corresponding permutation of  Z / p³Z.  More generally, we prove similar results about subsets of  Z / pⁿZ  which are permuted by  g.  For instance, if  g  transitively permutes a subset  S  of  Z / pⁿZ  with  n>2,  and if reduction mod  pⁿ-1  is a  p-to-one map on  S,  then  g  transitively permutes the  p^k #S  elements of  Z / p^n+kZ  which reduce mod  pⁿ  to elements of  S. 

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