Translation planes of large dimension admitting nonsolvable groups
type: Item
instance of: scholarly article
title: Translation planes of large dimension admitting nonsolvable groups
zbMATH Open document ID: 0766.51005
DOI: 10.1007/BF01225768
author: Vikram Jha
author: Norman L. Johnson
published in: Journal of Geometry
publication date: 1993-01-17T00:00:00Z
review text: Authors look for (finite) translation planes with large Ostrom dimension admitting nonsolvable collineation groups. Using the technique of spreads and irreducible pairs (representation of a translation plane in vector space over its kernel) the authors show that the spread of order \(q^ 3\) associated with an irreducible pair on an \(3m\)-dimensional vector space over \(GF(p)\), \(q=p^ m\), admits \(GL(2,q)\) with sufficiently regular action [Cor. (4.5)]. Explicit examples of irreducible pairs [Cor. (6.2) and Thm. (6.3)] yield the main result: for any integer \(N\) there exist a translation plane of dimension \(>N\) and order \(q^ 3\) admitting \(GL(2,q)\) as a collineation group. Known examples of planes with small kernels, due to Kantor and to Bartolone and Ostrom can be obtained following the technique proposed in the paper.
Mathematics Subject Classification ID: 51A40
zbMATH DE Number: 98731
zbMATH Keywords: semifield
zbMATH Keywords: translation planes
zbMATH Keywords: spreads
zbMATH Keywords: irreducible pairs
reviewed by: K. Prażmowski
MaRDI profile type: Publication
cites work: Translation planes of order \(q^ 3\) which admit SL(2,q)
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cites work: Elementary Abelian 2-groups in finite translation planes
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description: scientific article; zbMATH DE number 98731