ON THE NUMBER OF QUADRATIC ORTHOMORPHISMS THAT PRODUCE MAXIMALLY NONASSOCIATIVE QUASIGROUPS

Abstract

Abstract Let q be an odd prime power and suppose that $a,b\in \mathbb {F}_q$ are such that $ab$ and $(1{-}a)(1{-}b)$ are nonzero squares. Let $Q_{a,b} = (\mathbb {F}_q,*)$ be the quasigroup in which the operation is defined by $u*v=u+a(v{-}u)$ if $v-u$ is a square, and $u*v=u+b(v{-}u)$ if $v-u$ is a nonsquare. This quasigroup is called maximally nonassociative if it satisfies $x*(y*z) = (x*y)*z \Leftrightarrow x=y=z$ . Denote by $\sigma (q)$ the number of $(a,b)$ for which $Q_{a,b}$ is maximally nonassociative. We show that there exist constants $\alpha \approx 0.029\,08$ and $\beta \approx 0.012\,59$ such that if $q\equiv 1 \bmod 4$ , then $\lim \sigma (q)/q^2 = \alpha $ , and if $q \equiv 3 \bmod 4$ , then $\lim \sigma (q)/q^2 = \beta $ .