The derivation of dual translation planes
type: Item
instance of: scholarly article
title: The derivation of dual translation planes
zbMATH Open document ID: 0695.51003
DOI: 10.1007/BF01231023
published in: Journal of Geometry
publication date: 1989-01-01T00:00:00Z
review text: Let \(\Pi\) be a finite affine plane of order n with a derivable net R, and let \(\Pi^*\) be the plane obtained from \(\Pi\) by deriving R. The group of collineations of \(\Pi\) which map the derivable net R onto itself induces a collineation group on \(\Pi^*\) called the inherited group of \(\Pi^*\). Question: Is the full collineation group of \(\Pi^*\) the inherited group? In general the answer is no; for example, the Hall translation planes are derivable and their derived planes are the desarguesian planes. Consider this question under the additional conditions: (1) the plane \(\Pi\) is a dual translation plane of order \(q^ 2>16\), and (2) the derivable net contains a shears axis. For the known planes satisfying these conditions the answer is yes to the above question. However, if \(\Pi\) is the semifield plane of order 16 with kernel GF(2) and \(\Pi^*\) is the Dempwolff plane of order 16, then the answer is no. The author proves that under the added conditions either the answer is yes or \(\Pi\) and \(\Pi^*\) satisfy certain properties including: (a) q is even, and (b) \(\Pi\) has translations which are not dual translations. (For the full list, see Theorem A of the article. These conditions are satisfied by the Dempwolff plane, but whether there exists a plane of order \(q^ 2>16\) satisfying them is not known.) If \(\Pi_ 1\) and \(\Pi_ 2\) are two dual translation planes with derivable nets \(R_ 1\) and \(R_ 2\) respectively both satisfying conditions (1) and (2), then the article's second major result is that \(\Pi_ 1\) not isomorphic to \(\Pi_ 2\) implies the corresponding derived planes are not isomorphic. The second major thrust of this article is the investigation of the number of non-isomorphic derived planes obtained from the same plane \(\Pi\). The main result in this direction handles the case where the derivable net R is a kernel net. To define kernel net, let \(\Pi\) be a dual translation plane of order \(q^ 2\), and let \(\Pi_ 0\) be the correspond…
reviewed by: Michael J. Kallaher
Mathematics Subject Classification ID: 51A40
Mathematics Subject Classification ID: 51E15
zbMATH DE Number: 4138656
zbMATH Keywords: derivation
zbMATH Keywords: inherited group
zbMATH Keywords: dual translation plane
zbMATH Keywords: kernel net
author: Norman L. Johnson
MaRDI profile type: Publication
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full work available at URL: https://doi.org/10.1007/bf01231023
OpenAlex ID: W1998279854
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description: scientific article; zbMATH DE number 4138656