Maximal nonassociativity via nearfields

摘要

We say that (x, y, z) is an element of Q(3) is an associative triple in a quasigroup Q(*) if (x * y) * z = x * (y * z). It is easy to show that the number of associative triples in Q is at least vertical bar Q vertical bar and it was conjectured that quasigroups with exactly vertical bar Q vertical bar associative triples do not exist when vertical bar Q vertical bar 1. We refute this conjecture by proving the existence of quasigroups with exactly vertical bar Q vertical bar associative triples for a wide range of values vertical bar Q vertical bar. Our main tools are quadratic Dickson nearfields and the Weil bound on quadratic character sums. (C) 2019 Elsevier Inc. All rights reserved.