Groups of Polynomial Permutations over Finite Commutative Rings

Abstract:

Sophie Frisch characterized the structure of the group of polynomial permutations over $\mathbb{Z}/p^2\mathbb{Z}$. Qifan Zhang found a correspondence between polynomial functions over $\mathbb{Z}/p^2\mathbb{Z}$ and 3-tuples of polynomial functions over $\mathbb{Z}/p\mathbb{Z}$, this paper is giving another proof of [1]. In this paper, we first prove that over any finite commutative ring $R$, the group of polynomial permutations is isomorphic to the automorphism group of the $R$-algebra of the polynomial functions. Then we give an easy proof to the characterization of Sophie Frisch using the correspondence set found by Zhang.