A strengthening of McConnel's theorem on permutations over finite fields

Abstract

Let $p$ be a prime, $q=p^n$, and $D\subset {\mathbb{F}}_q^{\ast }$. A celebrated result of McConnel states that if $D$ is a proper subgroup of ${\mathbb{F}}_q^{\ast }$, and $f:{\mathbb{F}}_q\to {\mathbb{F}}_q$ is a function such that $(f(x)-f(y))/(x-y)\in D$ whenever $x\ne y$, then $f(x)$ necessarily has the form $a{x}^{p^j}+b$. In this notes, we give a sufficient condition on $D$ to obtain the same conclusion on $f$. In particular, we show that McConnel's theorem extends if $D$ has small doubling.

arXiv comment5 pages, revised based on referee comments

References in corpus (5)

• A Carlitz type result for linearized polynomials
• Van Lint-MacWilliams' conjecture and maximum cliques in Cayley graphs over finite fields
• Permutation groups of prime degree, a quick proof of Burnside's theorem
• Erdős-Ko-Rado theorem in Peisert-type graphs
• Intersecting families of graphs of functions over a finite field