Maximal nonassociativity via fields

Abstract

We say that $(x,y,z)\in Q^3$ is an associative triple in a quasigroup $(Q,*)$ if $(x*y)*z=x*(y*z)$. Let a(Q) denote the number of associative triples in Q. It is easy to show that $a(Q)\ge |Q|$, and we call the quasigroup maximally nonassociative if $a(Q)= |Q|$. It was conjectured that maximally nonassociative quasigroups do not exist when $|Q|>1$. Drápal and Lisoněk recently refuted this conjecture by proving the existence of maximally nonassociative quasigroups for a certain infinite set of orders |Q|. In this paper we prove the existence of maximally nonassociative quasigroups for a much larger set of orders |Q|. Our main tools are finite fields and the Weil bound on quadratic character sums. Unlike in the previous work, our results are to a large extent constructive.