Few associative triples, isotopisms and groups

Abstract

Let Q be a quasigroup. For $$\alpha ,\beta \in S_Q$$ let $$Q_{\alpha ,\beta }$$ be the principal isotope $$x*y = \alpha (x)\beta (y)$$ . Put $$\mathbf a(Q)= |\{(x,y,z)\in Q^3;$$ $$x(yz)) = (xy)z\}|$$ and assume that $$|Q|=n$$ . Then $$\sum _{\alpha ,\beta }\mathbf a(Q_{\alpha ,\beta })/(n!)^2 = n^2(1+(n-1)^{-1})$$ , and for every $$\alpha \in S_Q$$ there is $$\sum _\beta \mathbf a(Q_{\alpha ,\beta })/n! = n(n-1)^{-1}\sum _x(f_x^2-2f_x+n)\ge n^2$$ , where $$f_x=|\{y\in Q;$$ $$ y = \alpha (y)x\}|$$ . If G is a group and $$\alpha $$ is an orthomorphism, then $$\mathbf a(G_{\alpha ,\beta })=n^2$$ for every $$\beta \in S_Q$$ . A detailed case study of $$\mathbf a(G_{\alpha ,\beta })$$ is made for the situation when $$G = \mathbb Z_{2d}$$ , and both $$\alpha $$ and $$\beta $$ are “natural” near-orthomorphisms. Asymptotically, $$\mathbf a(G_{\alpha ,\beta })>3n$$ if G is an abelian group of order n. Computational results: $$\mathbf a(7) = 17$$ and $$\mathbf a(8) \le 21$$ , where $$\mathbf a(n) = \min \{\mathbf a(Q);$$ $$ |Q|=n\}$$ . There are also determined minimum values for $$\mathbf a(G_{\alpha ,\beta })$$ , G a group of order $$\le 8$$ .