Transitive projective planes and 2-rank

TL;DR

This paper investigates the symmetry properties of non-Desarguesian projective planes under group actions, revealing constraints on Sylow 2-subgroups and automorphism groups, which deepen understanding of their algebraic structure.

Contribution

It establishes that Sylow 2-subgroups are cyclic or quaternionic and that such planes have an odd order automorphism group acting transitively, advancing the classification of these geometries.

Findings

Sylow 2-subgroups are cyclic or generalized quaternion

Existence of an odd order automorphism group acting transitively

Constraints on the symmetry groups of non-Desarguesian planes

Abstract

Suppose that a group $G$ acts transitively on the points of a non-Desarguesian plane, $\mathcal{P}$. We prove first that the Sylow 2-subgroups of $G$ are cyclic or generalized quaternion. We also prove that $\mathcal{P}$ must admit an odd order automorphism group which acts transitively on the set of points of $\mathcal{P}$.