Cycle structures of autotopisms of the Latin squares of order up to 11

TL;DR

This paper classifies all autotopisms of Latin squares up to order 11 by analyzing their cycle structures, providing insights into their symmetry properties.

Contribution

It offers a comprehensive classification of autotopisms for Latin squares of order up to 11 based on their cycle structures, a novel detailed analysis.

Findings

Complete classification of autotopisms up to order 11

Identification of cycle structure patterns in autotopisms

Insights into symmetry properties of Latin squares

Abstract

The cycle structure of a Latin square autotopism $\Theta=(\alpha,\beta,\gamma)$ is the triple $(\mathbf{l}_{\alpha},\mathbf{l}_{\beta},\mathbf{l}_{\gamma})$, where $\mathbf{l}_{\delta}$ is the cycle structure of $\delta$, for all $\delta\in\{\alpha,\beta,\gamma\}$. In this paper we study some properties of these cycle structures and, as a consequence, we give a classification of all autotopisms of the Latin squares of order up to 11.