Transitive latin bitrades

TL;DR

This paper investigates properties of latin bitrades, establishing conditions under which they derive from group-based constructions and analyzing their genus 0 case for uniqueness and autotopism group relations.

Contribution

It provides new results linking primary, thin, separated latin bitrades with group-based constructions and characterizes genus 0 latin bitrades for uniqueness and autotopism group equality.

Findings

Primary, thin, separated latin bitrades with regular autotopism groups derive from groups.

Genus 0 latin bitrades have unique disjoint mates.

Autotopism groups of genus 0 latin bitrades are equal for both mates.

Abstract

In this note we give two results. First, if a latin bitrade $(T_1, T_2)$ is primary, thin, separated, and the autotopism group of $T_1$ acts regularly on $T_1$, then $(T_1, T_2)$ may be derived from a group-based construction. Second, if a latin bitrade $(T_1, T_2)$ has genus 0 then the disjoint mate $T_2$ is unique and the autotopism group of $T_1$ is equal to the autotopism group of $T_2$.