Maximal nonassociativity via nearfields (Q2302570)

For a finite quasigroup \(Q\), we let \(a(Q) := |\{ (x,y,z) \in Q^3 \,:\, (xy)z = x(yz)\}|\) be the number of associative triples of \(Q\). \textit{O. Grošek} and \textit{P. Horák} [Des. Codes Cryptography 64, No. 1--2, 221--227 (2012; Zbl 1250.94036)] proved that \(a(Q) \ge 2 |Q| - i(Q)\), where \(i(Q)\) denotes the number of idempotent elements of \(Q\) and conjectured that \(a (Q) > |Q|\) for all \(Q\) (with \(|Q| > 1\)). The paper under review refutes this conjecture by proving that for every \(k \ge 0\) and every odd number \(r\), there is a quasigroup \(Q\) with \(a (Q) = |Q| = 2^{6k} r^2\); a quasigroup with \(a (Q) = |Q|\) is called \textit{maximally nonassociative}. Such quasigroups are constructed as direct products of quasigroups obtained from finite (left) Dickson nearfields. For an odd prime power \(q\), the nearfield \(N_{q^2} : = (\mathbb{F}_{q^2}, +,\circ)\) is defined by \(x \circ y = xy\) if \(x\) is a square in \(\mathbb{F}_{q^2}\) and \(x \circ y = xy^q\) otherwise. Then for \(c \in N_{q^2} \setminus \{0,1\}\), an operation \(*_c\) is defined by \(x*_c y := x + (y-x) \circ c\). The authors show that if \(q \ge 14293\), there is a \(c\) such such that \((N_{q^2}, *_c)\) is maximally nonassociative by an involved estimation of the number of solutions of certain polynomial equations using the Weil bound on multiplicative character sums; smaller \(q\) are investigated using a computer algebra system. Computer generated results indicate that for \(q \to \infty\), the density of those \(c\) such that \((N_{q^2}, *_c)\) is maximally nonassociative converges to a number in the interval \((0.288, 0.290)\) (Conjecture~5.10).