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11.02.2017 | Ausgabe 3/2018

# Few associative triples, isotopisms and groups

Zeitschrift:
Designs, Codes and Cryptography > Ausgabe 3/2018
Autoren:
Aleš Drápal, Viliam Valent
Wichtige Hinweise
Communicated by D. Jungnickel.

## 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 $$\le 8$$.

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