On Buekenhout-Metz unitals in \(PG(2, q^ 2)\), \(q\) even
type: Item
instance of: scholarly article
title: On Buekenhout-Metz unitals in \(PG(2, q^ 2)\), \(q\) even
zbMATH Open document ID: 0759.51009
DOI: 10.1007/BF01190683
author: Vito Abatangelo
published in: Archiv der Mathematik
publication date: 1993-02-10T00:00:00Z
review text: The classical example of a Hermitian arc (also called an embedded unital) in \(PG(2,q^ 2)\) is a non-singular Hermitian curve. Using an idea of F. Buekenhout, R. Metz exhibited some non-classical unitals which are known as parabolic Buekenhout-Metz unitals, or briefly parabolic BM unitals [\textit{R. Metz}, Geom. Dedicata 8, 125-126 (1979; Zbl 0403.51007)]; for an alternative approach see [\textit{L. Maria Abatangelo}, Matematiche 39, No. 1/2, 101-110 (1984; 632.51005)]. So for collineation groups fixing parabolic BM unitals have been studied in the odd order case [cf. the author and \textit{B. Larato}, Boll. Unione Mat. Ital., VII. Ser., A5, No. 2, 195-206 (1991; Zbl 0746.51013) and \textit{R. D. Baker} and \textit{G. L. Ebert}, On Buekenhout-Metz unitals of odd order, preprint; see also J. Comb. Theory, Ser. A60, No. 1, 67-84 (1992)]. In this paper we show that the linear collineation group \(\Gamma\) fixing a parabolic BM unital \({\mathcal U}\) of \(PG(2,q^ 2)\), \(q\) even, has the following properties: (1.1) \(\Gamma\) fixes the special point \(A\) of \(\mathcal U\). (1.2) \(\Gamma\) has a normal subgroup \(S\) acting on the non-special points of \(\mathcal U\) as a sharply transitive group. (1.3) The one-point stabilizer in \(\Gamma\) has order \(q-1\) for any non- special point of \(\mathcal U\). The main result is that the above properties characterize the parabolic BM unitals among the non-classical unitals of \(PG(2,q^ 2)\), \(q\) even.
reviewed by: Vito Abatangelo
Mathematics Subject Classification ID: 51E30
zbMATH DE Number: 119850
zbMATH Keywords: Hermitian arc
zbMATH Keywords: unital
zbMATH Keywords: parabolic Buekenhout-Metz unitals
zbMATH Keywords: 633
zbMATH Keywords: linear collineation group
MaRDI profile type: Publication
cites work: Q3984383
cites work: On Buekenhout-Metz unitals of odd order
cites work: Collineation groups preserving a unital in a projective plane of even order
cites work: Existence of unitals in finite translation planes of order \(q^2\) with a kernel of order \(q\)
cites work: Q3694152
cites work: On Unitary Polarities of Finite Projective Planes
cites work: On a class of unitals
full work available at URL: https://doi.org/10.1007/bf01190683
OpenAlex ID: W2057950455
Recommended article: Q3769501
Recommended article: Unitals which meet Baer subplanes in 1 modulo \(q\) points
Recommended article: A characterization of Buekenhout-Metz unitals in PG\((2,q^ 2)\), \(q\) even
Recommended article: Q3984383
Recommended article: Unitals in PG\((2,q^2)\) with a large 2-point stabiliser
Recommended article: Q4218477
Recommended article: Unitals in finite Desarguesian planes
Recommended article: Intersections of Buekenhout-Metz unitals
Recommended article: A group theoretic characterization of Buekenhout-Metz unitals in PG\((2,q^2)\) containing conics
Recommended article: Group theoretic characterizations of Buekenhout-Metz unitals in PG\((2,q^{2})\)
description: scientific article; zbMATH DE number 119850