Collineation groups preserving a unital in a projective plane of even order
type: Item
instance of: scholarly article
title: Collineation groups preserving a unital in a projective plane of even order
zbMATH Open document ID: 0694.51008
DOI: 10.1007/BF00147464
author: Mauro Biliotti
author: Gabor Korchmáros
published in: Geometriae Dedicata
publication date: 1989-01-01T00:00:00Z
review text: Let \(\Pi\) be a finite projective plane of order \(m^ 2\); a unital \({\mathcal U}\) embedded in \(\Pi\) is a substructure of \(\Pi\) which forms a \(2-(m^ 3+1,m+1,1)\) design, and in particular a ``classical'' unital consists of absolute points and nonabsolute lines of a Hermitian polarity in a Desarguesian plane of order \(m^ 2.\) In this work the authors prove that, if \(\Pi\) is of even order \(m^ 2\) and admits a collineation group G which preserves a unital \({\mathcal U}\), such that G is transitive on the points of \({\mathcal U}\) and the socle of G has even order, then G contains involutory elations: in this case \(\Pi\) is Desarguesian, \({\mathcal U}\) is classical and if \(m>2\), \(PSU(3,m^ 2)\leq G\leq P\Gamma U(3,m^ 2).\) In this way as corollary the authors obtain that, if G is primitive on the points of \({\mathcal U}\), then \(\Pi\) is Desarguesian, \({\mathcal U}\) is classical and, if \(m>2\), \(PSU(3,m^ 2)\leq G\leq P\Gamma U(3,m^ 2)\).
reviewed by: V.Dicuonzo
Mathematics Subject Classification ID: 51E20
Mathematics Subject Classification ID: 51E15
zbMATH DE Number: 4136872
zbMATH Keywords: projective plane of even order
zbMATH Keywords: collineation
zbMATH Keywords: unital
Wikidata QID: Q59249295
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 4136872