A note on the group of projectivities of finite projective planes.
type: Item
instance of: scholarly article
title: A note on the group of projectivities of finite projective planes.
zbMATH Open document ID: 1172.51007
author: Peter Müller
author: G. P. Nagy
published in: Innovations in Incidence Geometry
publication date: 2009-06-23T00:00:00Z
review text: In his paper [J. Geom. 20, 72--85 (1983; Zbl 0514.51013)], \textit{T. Grundhöfer} has shown that the group of projectivities of a non-Desarguesian projective plane of finite order \(n\) is either the alternating group \(A_{n+1}\), or the symmetric group \(S_{n+1}\), or \(n=23\), and the group of projectivities in this case is the Mathieu group \(M_{24}\). In the present paper the authors show that the latter case cannot occur. A crucial step in their argument is the observation that the Mathieu group \(M_{22}\) does not contain the multiplication group of loop order 22. (For a slightly weaker result, see [\textit{A. Drápal}, J. Algebra 251, No.~1, 256--278 (2002; Zbl 1009.20079)].) To prove this, the authors use also computer calculation.
reviewed by: József Szilasi
Mathematics Subject Classification ID: 51E15
Mathematics Subject Classification ID: 20N05
zbMATH DE Number: 5567868
zbMATH Keywords: projective planes
zbMATH Keywords: projectivities
zbMATH Keywords: loops
MaRDI profile type: Publication
full work available at URL: iig.2008.6.291
OpenAlex ID: W2915178302
Wikidata QID: Q128361980
DOI: 10.2140/IIG.2008.6.291
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description: scientific article; zbMATH DE number 5567868