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I'm wondering, which is the smallest quasigroup, which is not a group? And how to check it?

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Let us say "which are the smallest quasigroups"! –  MattAllegro Apr 18 at 23:00

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Looking at Wikipedia: Small Latin squares and quasigroups it is quite clear that quasigroups of order two or below are really groups, while for order three there is a quasigroup with no identity element.

(Of course if you allow the empty set with its unique binary opeation as a quasigroup, that structure is not a group.)

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The Cayley table:

$$\begin{array}{c|ccc} \ast & 0 & 1 & 2 \\ \hline 0 & 0 & 2 & 1 \\ 1 & 1 & 0 & 2 \\ 2 & 2 & 1 & 0 \end{array}$$ represents a finite quasigroup of order $3$ on the set $\mathbb{Z}_3$ of the integers mod$3$. The operation $\ast$ is $$a\ast b=(a-b)\text{mod}3.$$


Check that the operations $$a\bullet b=(a+b)\text{mod}2$$ and $$a\circ b=(a-b)\text{mod}2$$ on the set $\mathbb{Z}_2$ of the integers mod$2$ give raise to the same Cayley table: $$\begin{array}{c|cc} \bullet & 0 & 1 &\\ \hline 0 & 0 & 1 \\ 1 & 1 & 0 \end{array}$$

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