Transitive projective planes and insoluble groups

TL;DR

This paper investigates the structure of groups acting transitively on points of finite non-Desarguesian projective planes, showing that insoluble groups have a very specific quotient structure related to SL_2(5).

Contribution

It proves that insoluble groups acting transitively on such planes have quotients isomorphic to SL_2(5) or its extension, revealing a precise structural constraint.

Findings

Insoluble groups have quotients isomorphic to SL_2(5) or SL_2(5).2.

Provides structural classification for groups acting on non-Desarguesian planes.

Abstract

Suppose that a group $G$ acts transitively on the points of $\mathcal{P}$, a finite non-Desarguesian projective plane. We prove that if $G$ is insoluble then $G/O(G)$ is isomorphic to $SL_2(5)$ or $SL_2(5).2$.