On a theorem of Rigby
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 \(q\).
scientific article; zbMATH DE number 6680045
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