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 $\mathrm{Par}(n)$ 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 $\mathrm{Par}(n)$ for $n\le17$. We also study the proportion of all paratopisms that are in $\mathrm{Par}(n)$ as $n\rightarrow\infty$.