Autoparatopisms of Quasigroups and Latin Squares

Paratopism is a well known action of the wreath product ${\mathcal{S}}_n\wr {\mathcal{S}}_3$ on Latin squares of order $n$. A paratopism that maps a Latin square to itself is an autoparatopism of that Latin square. Let $\mathrm{P}\mathrm{a}\mathrm{r}(n)$ denote the set of paratopisms that are an autoparatopism of at least one Latin square of order $n$. We prove a number of general properties of autoparatopisms. Applying these results, we determine $\mathrm{P}\mathrm{a}\mathrm{r}(n)$ for $n\le 17$. We also study the proportion of all paratopisms that are in $\mathrm{P}\mathrm{a}\mathrm{r}(n)$ as $n\to \mathrm{\infty }$.