Transitive projective planes

TL;DR

This paper investigates the longstanding conjecture that all transitive finite projective planes are Desarguesian, providing evidence that groups acting transitively on non-Desarguesian planes lack components.

Contribution

It demonstrates that a transitive group on a non-Desarguesian projective plane cannot contain any components, advancing understanding towards the conjecture.

Findings

Groups acting transitively on non-Desarguesian planes have no components

Supports the conjecture that all transitive finite projective planes are Desarguesian

Provides structural constraints on symmetry groups of such planes

Abstract

A long-standing conjecture is that any transitive finite projective plane is Desarguesian. We make a contribution towards a proof of this conjecture by showing that a group acting transitively on the the points of a non-Desarguesianprojective plane must not contain any components.