Linear groups as right multiplication groups of quasifields

TL;DR

This paper classifies finite quasifields based on their right multiplication groups, focusing on those with exceptional transitive linear groups, and explores the relationships between parastrophy and isotopy.

Contribution

It provides a classification of quasifields with exceptional transitive linear groups as their right multiplication groups, linking parastrophy to isomorphism of translation planes.

Findings

Classified quasifields with specific linear groups

Established equivalence between parastrophy and isomorphism of translation planes

Identified all quasifields with exceptional finite transitive linear groups

Abstract

For quasifields, the concept of parastrophy is slightly weaker than isotopy. Parastrophic quasifields yield isomorphic translation planes but not conversely. We investigate the right multiplication groups of finite quasifields. We classify all quasifields having an exceptional finite transitive linear group as right multiplication group. The classification is up to parastrophy, which turns out to be the same as up to the isomorphism of the corresponding translation planes.