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 $|Q|>1$. Dr\'apal and Lison\v{e}k 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.