ON THE NUMBER OF QUADRATIC ORTHOMORPHISMS THAT PRODUCE MAXIMALLY NONASSOCIATIVE QUASIGROUPS

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