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 → K that agree with an F p -affine function K → K on each coset of a given additive subgroup W of K -we call them W -coset-wise 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.