(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 pk #S elements of Z / pn+kZ which reduce mod pn to elements of S.
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