On planes of order \(p^2\) in which every quadrangle generates a subplane of order \(p\)
type: Item
instance of: scholarly article
title: On planes of order \(p^2\) in which every quadrangle generates a subplane of order \(p\)
zbMATH Open document ID: 0969.51002
DOI: 10.1023/A:1005263604160
author: Aart Blokhuis
author: Peter Sziklai
published in: Geometriae Dedicata
publication date: 2001-10-02T00:00:00Z
review text: According to \textit{A. M. Gleason} [Am. J. Math. 78, 797-807 (1956; Zbl 0072.38001)], a finite projective plane is desarguesian if every quadrangle is contained in a subplane of order 2. The authors prove the following beautiful result: let \(p\) be a prime; if every quadrangle of a finite projective plane of order \(p^2\) is contained in a subplane of order \(p\), then the plane is desarguesian. The proof relies on a theorem of \textit{J. Cofman} [Geom. Dedicata 2, 195-199 (1973; Zbl 0279.50015)] and on the enumeration of all doubly transitive permutation groups of degree \(p^2\) (which seems to require the classification of all finite simple groups).
reviewed by: Theo Grundhöfer
Mathematics Subject Classification ID: 51A30
Mathematics Subject Classification ID: 51E12
Mathematics Subject Classification ID: 51E15
zbMATH DE Number: 1455309
zbMATH Keywords: Baer subplane
zbMATH Keywords: projective plane
zbMATH Keywords: desarguesian
zbMATH Keywords: quadrangle
MaRDI profile type: Publication
full work available at URL: a:1005263604160
OpenAlex ID: W1821235970
description: scientific article; zbMATH DE number 1455309