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ON THE NUMBER OF QUADRATIC ORTHOMORPHISMS THAT PRODUCE MAXIMALLY NONASSOCIATIVE QUASIGROUPS
- Part of
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- Journal:
- Journal of the Australian Mathematical Society / Volume 115 / Issue 3 / December 2023
- Published online by Cambridge University Press:
- 20 February 2023, pp. 311-336
- Print publication:
- December 2023
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- Article
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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 $.
