On a theorem of Rigby
type: Item
instance of: scholarly article
title: On a theorem of Rigby
zbMATH Open document ID: 1360.51003
author: Dieter Jungnickel
published in: Journal of Geometry
publication date: 2017-02-02T00:00:00Z
review text: In the present paper, the author studies a natural question whether certain (symmetric) configurations can be embedded into the projective plane \(\Pi =\mathrm{PG}(2,F)\), where \(F\) is a (not necessarily commutative) field. The main object of the paper is the classical biaffine plane of order \(q\), denoted by \(\Delta=\mathrm{BAG}(2,q)\), that is, the symmetric configuration \(((q^{2}-1)_{q})\) obtained from some classical affine plane \(\Sigma = \mathrm{AG}(2,q)\) by ommiting a point of \(\Sigma\) together with all lines passing through this point. The main result of this note is a variation (and strengthening) on a result due to \textit{Rigby} [Can. J. Math. 17, 977--1009 (1965; Zbl 0131.19201)]. Theorem 1. (Rigby). Let \(\Pi\) be a Desarguesian projective plane \(\mathrm{PG}(2,F)\) containing an affine subplane \(\Sigma\) of order \(q\geq 4\). Then, \(\Sigma\) is likewise Desarguesian (\(q\) is a power of a prime \(p\)), and \(\Pi\) contains the projective extension \(\mathrm{PG}(2,q)\) of \(\Sigma\) (so that \(F\) has characteristic \(p\)). In particular, \(\mathrm{AG}(2,q)\) with \(q\geq 4\) admits an embedding into \((\mathrm{PG}(2,q')\) if and only if \(q'\) is a power of \(q\). Main Result. Assume that a classical biaffine plane \(\Delta=\mathrm{BAG}(2,q)\) admits an embedding into the projective plane \(\Pi =\mathrm{PG}(2,F)\), where \(q\geq 4\) is a power of a prime \(p\) and \(F\) is a (not necessarily commutative) field. Then, the embedding of \(\triangle\) extends to a projective subplane \(\Pi_{0} \simeq\mathrm{PG}(2,q)\) of \(\Pi\) (so that \(F\) has characteristic \(p\)). An immediate corollary from the main result is the following. Corollary. The classical biaffine plane \(\mathrm{BAG}(2,q)\) with \(q \geq 4\) admits an embedding into \(\mathrm{PG}(2,q')\) if and only if \(q'\) is a power of \(q\).
reviewed by: Piotr Pokora
Mathematics Subject Classification ID: 51A45
Mathematics Subject Classification ID: 05B30
zbMATH DE Number: 6680045
zbMATH Keywords: configuration
zbMATH Keywords: projective plane
zbMATH Keywords: embedding theorems
MaRDI profile type: Publication
full work available at URL: s00022-015-0298-7
OpenAlex ID: W2206164252
cites work: \(8_3\) in \(\mathrm{PG}(2,q)\)
cites work: The geometric dimension of some small configurations
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cites work: Q3227201
cites work: Near-embeddings of the affine plane with 9 points into Desarguesian projective and affine planes
DOI: 10.1007/S00022-015-0298-7
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description: scientific article; zbMATH DE number 6680045