Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Autoparatopisms of Quasigroups and Latin Squares

Mahamendige Jayama Lalani Mendis, Ian M. Wanless

Abstract

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 $n\rightarrow\infty$.

Autoparatopisms of Quasigroups and Latin Squares

Abstract

Paratopism is a well known action of the wreath product on Latin squares of order . A paratopism that maps a Latin square to itself is an autoparatopism of that Latin square. Let denote the set of paratopisms that are an autoparatopism of at least one Latin square of order . We prove a number of general properties of autoparatopisms. Applying these results, we determine for . We also study the proportion of all paratopisms that are in as .
Paper Structure (4 sections, 4 theorems, 5 equations)

This paper contains 4 sections, 4 theorems, 5 equations.

Table of Contents

  1. Introduction
  2. Some basic tools and terminology
  3. Cycle structures
  4. Cell orbits

Key Result

Lemma 2.1

Suppose $\sigma_1$ and $\sigma_2$ are conjugate in $\mathcal{P}_n$. Then $\sigma_1 \in \mathrm{Par}(n)$ if and only if $\sigma_2 \in \mathrm{Par}(n)$.

Theorems & Definitions (6)

  • Lemma 2.1
  • Theorem 2.2
  • Lemma 2.3
  • proof
  • Lemma 2.4
  • proof