Cycle structure of autotopisms of quasigroups and Latin squares

TL;DR

This paper investigates the cycle structures of autotopisms of Latin squares, establishing necessary conditions, determining autotopisms for small orders, and characterizing automorphisms of quasigroups under specific cycle constraints.

Contribution

It provides new necessary conditions for autotopisms based on cycle structures and fully determines autotopisms for Latin squares of order up to 17, also characterizing automorphisms under certain cycle conditions.

Findings

Determined autotopisms for Latin squares of order up to 17.

Established necessary cycle structure conditions for autotopisms.

Characterized when a permutation is an automorphism of a quasigroup under specific cycle constraints.

Abstract

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