Translationsebenen der Ordnung 49 mit einer Quaternionengruppe von Dehnungen. (Translation planes of order 49 with a quaternion group of homologies)
type: Item
instance of: scholarly article
title: Translationsebenen der Ordnung 49 mit einer Quaternionengruppe von Dehnungen. (Translation planes of order 49 with a quaternion group of homologies)
zbMATH Open document ID: 0766.51004
DOI: 10.1007/BF01228283
author: Günter Heimbeck
published in: Journal of Geometry
publication date: 1993-01-16T00:00:00Z
review text: The author shows that there exist, up to isomorphism, exactly ten translation planes of order 49. He solves this problem by finding all systems \(\Sigma\subset GL_ 2(7)\) satisfying the following two conditions 1) for two vectors \(x,y\neq 0\) from \(GF(7)^ 2\) there exists exactly one \(\sigma\in\Sigma\) such that \(\sigma x=y\); 2) \(\Sigma\) contains a group of quaternions \(\Sigma_ 0\) such that \(\Sigma\Sigma_ 0=\Sigma\).
reviewed by: H.Makiewiecka
Mathematics Subject Classification ID: 51A40
Mathematics Subject Classification ID: 20G20
zbMATH DE Number: 92574
zbMATH Keywords: quaternion groups
zbMATH Keywords: vectorial spaces
zbMATH Keywords: translation planes of order 49
MaRDI profile type: Publication
cites work: Some translation planes of order \(p^ 2\) and of extra-special type
cites work: Q3704520
cites work: A new class of finite translation planes obtained from the exceptional near-fields
full work available at URL: https://doi.org/10.1007/bf01228283
OpenAlex ID: W2106454451
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description: scientific article; zbMATH DE number 92574