\(8_3\) in \(\mathrm{PG}(2,q)\)
type: Item
instance of: scholarly article
title: \(8_3\) in \(\mathrm{PG}(2,q)\)
zbMATH Open document ID: 0595.51003
DOI: 10.1007/BF01200478
author: M. S. Abdul-Elah
author: M. W. Al-Dhahir
author: Dieter Jungnickel
published in: Archiv der Mathematik
publication date: 1987-01-01T00:00:00Z
review text: Let \(\mathfrak C\) be a tactical configuration \(8_ 3\) embedded in a projective plane \({\mathfrak P}\). Then \(\mathfrak C\) is in fact isomorphic to \(\mathrm{BAG}(2,3)\), the biaffine plane of order 3 obtained by puncturing the affine plane of order 3. The main result is as follows: \(\mathfrak C\) can be embedded into a Pappian plane \(\mathfrak P=\mathrm{PG}(2,F)\) iff one of the following three cases holds: (i) \(\mathrm{char}\, F=3\); (ii) \(\mathrm{char}\, F=2\) and \(F\) contains a primitive 3rd root of unity; (iii) \(F\) contains a primitive sixth root of unity. In the finite case, this means \(F=\mathrm{GF}(q)\) with \(q\) a power of 3 or 4 or with \(q\equiv 1\bmod 6\). We also show, that each \(8_3\) embedded in \(\mathrm{PG}(2,F)\) extends to an affine plane of order 3 embedded in \(\mathrm{PG}(2,F)\). Thus our result strengthens the theorem of Ostrom and Sherk on embedding \(\mathrm{AG}(2,3)\) into \(\mathrm{PG}(2,q)\) [see \textit{T. G. Ostrom} and \textit{F. A. Sherk}, Can. Math. Bull. 7, 549--559 (1964; Zbl 0125.38401)]. We also compute the number of configurations \(8_3\) (and of affine planes \(\mathrm{AG}(2,3))\) embedded in \(\mathrm{PG}(2,q)\). Particular attention is given to the cases \(q=3\) and \(q=4\).
Mathematics Subject Classification ID: 51A45
Mathematics Subject Classification ID: 51E15
zbMATH DE Number: 3958101
zbMATH Keywords: tactical configuration embedded in projective plane
zbMATH Keywords: punctured affine plane
MaRDI profile type: Publication
cites work: Q5589781
cites work: Q5573028
cites work: Q4160010
cites work: Q4370894
cites work: Finite Projective Planes with Affine Subplanes
cites work: Affine Subplanes of Finite Projective Planes
cites work: Q5808757
full work available at URL: https://doi.org/10.1007/bf01200478
OpenAlex ID: W2069472366
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description: scientific article; zbMATH DE number 3958101