Permutation polynomials over finite rings

Let PPol(R) denote the group of permutation polynomial functions over the finite, commutative, unital ring R under composition. We generalize numerous results about permutation polynomials over Z to local rings by treating them under a unified manner. In particular, we provide a natural wreath product decomposition of permutation polynomial functions over the maximal ideal M and over the finite field R/M. We characterize the group of permutation polynomial functions over M whenever the condition M-vertical bar R/M vertical bar = {0} applies. Then we derive the size of PPol(R), thereby generalizing the same size formulas for Z(p)(n). Finally, we completely characterize when the group PPol(R) is solvable, nilpotent, or abelian. (C) 2017 Elsevier Inc. All rights reserved.