On likeable translation planes of even order
type: Item
instance of: scholarly article
title: On likeable translation planes of even order
zbMATH Open document ID: 0534.51005
DOI: 10.1007/BF01207578
author: Michael J. Ganley
published in: Archiv der Mathematik
publication date: 1983-01-01T00:00:00Z
review text: A function f: GF(q)\(\to GF(q)\), where q is not a power of 3, is called ''likeable'' if (i) f is additive and (ii) the equation \(x^ 2=xy^ 2- \frac{1}{3}y^ 4+yf(y)\) has only \((x,y)=(0,0)\) as a solution. Using such a function one may construct a translation plane, which is also called likeable. The concept of a likeable function was introduced by W. M. Kantor [Isr. J. Math. 42, 227-234 (1982; Zbl 0511.51011)]. He observed that when \(q=2^ r\) (r odd, \(r\geq 3)\) the function \(f(y)=c^ 2y+cy^ 2 (c\in GF(2^ r))\) is likeable and that the planes obtained from different values of c are all isomorphic. The plane obtained is the so- called Betten-Walker plane of order \(2^{2r}\) [the reviewer, Geom. Dedicata 2, 327-339 (1973; Zbl 0272.50028), and \textit{M. Walker}, Geom. Dedicata 5, 135-146 (1976, Zbl 0356.50022)]. The author proves this Theorem: The only likeable translation plane of even order \(2^{2r}\) is the Betten-Walker plane, which occurs if and only if r is odd and \(r\geq 3\).
Mathematics Subject Classification ID: 51A40
Mathematics Subject Classification ID: 51E15
zbMATH DE Number: 3848117
zbMATH Keywords: Betten-Walker plane
zbMATH Keywords: spread
zbMATH Keywords: likeable function
MaRDI profile type: Publication
cites work: Commutative semifields, two dimensional over their middle nuclei
cites work: On point-transitive affine planes
cites work: Polynomials and linear transformations over finite fields.
full work available at URL: https://doi.org/10.1007/bf01207578
OpenAlex ID: W1991546586
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description: scientific article; zbMATH DE number 3848117