Coset-wise affine functions and cycle types of complete mappings

Abstract

Let $K$ be a finite field of characteristic $p$. We study a certain class of functions $K\to K$ that agree with an ${\mathbb{F}}_p$-affine function $K\to K$ on each coset of a given additive subgroup $W$ of $K$ - we call them $W$-coset-wise ${\mathbb{F}}_p$-affine functions of $K$. We show that these functions form a permutation group on $K$ with the structure of an imprimitive wreath product and characterize which of them are complete mappings of $K$. As a consequence, we are able to provide various new examples of cycle types of complete mappings of $K$, including that $K$ has a complete mapping moving all elements of $K$ in one cycle if $p>2$.

29 pages