Cycle structure of autotopisms of quasigroups and Latin squares

Abstract

An autotopism of a Latin square is a triple $(α,β,γ)$ of permutations such that the Latin square is mapped to itself by permuting its rows by $α$, columns by $β$, and symbols by $γ$. Let $\mathrm{Atp}(n)$ be the set of all autotopisms of Latin squares of order $n$. Whether a triple $(α,β,γ)$ of permutations belongs to $\mathrm{Atp}(n)$ depends only on the cycle structures of $α$, $β$ and $γ$. We establish a number of necessary conditions for $(α,β,γ)$ to be in $\mathrm{Atp}(n)$, and use them to determine $\mathrm{Atp}(n)$ for $n\le17$. For general $n$ we determine if $(α,α,α)\in\mathrm{Atp}(n)$ (that is, if $α$ is an automorphism of some quasigroup of order $n$), provided that either $α$ has at most three cycles other than fixed points or that the non-fixed points of $α$ are in cycles of the same length.