André planes and nests of reguli
type: Item
instance of: scholarly article
title: André planes and nests of reguli
zbMATH Open document ID: 0724.51013
DOI: 10.1007/BF00147458
author: Norman L. Johnson
author: Rolando J. Pomareda
published in: Geometriae Dedicata
publication date: 1989-01-01T00:00:00Z
review text: In this interesting paper, the authors determine all translation planes that admit certain collineation groups. Sometimes only the order of the group is considered. For instance, let \(\pi\) be a translation plane of order \(q^ n\) with kernel containing GF(q) that admits a linear collineation of order \([(q^ n-1)/(q-1)]^ 2.\) The authors show that if \[ ((q^ n-1)/(q-1),\prod^{n-1}_{i=1}(q^ i-1))=1, \] then \(\pi\) is an André plane. More typical of the results found in the paper are the following two: (1) Let \(\pi\) be a translation plane of order \(q^ n\), \(n>2\), with kernel GF(q). If \(\pi\) admits an abelian collineation group G of order \(((q^ n-1)/(q-1))^ 2\) in the linear translation complement, then \(\pi\) is a generalized André plane. (2) If \(\pi\) is a translation plane of order \(q^ n\) with kernel GF(q) that admits an abelian collineation group G of order \((q^ n-1)^ 2\) in the linear translation complement, then \(\pi\) is an André nearfield plane. The reviewer [Eur. J. Comb. 10, No.4, 319-330 (1989; Zbl 0691.05009)] constructed a family of translation planes of order \(q^ 2\) and kernel GF(q) by replacing a \((q+1)\)-nest of reguli. In general, a t-nest of reguli is a collection of t reguli in a regular spread so that every line of the spread is contained in zero or two reguli of the collection. In the paper under review the following result is proven: Let \(\pi\) be a translation plane of order \(q^ 2\) with kernel containing GF(q) that admits a collineation group G in the translation complement which is the direct product of two cyclic homology groups of order \(q+1\) with affine axes. Then either (i) \(\pi\) is André, (ii) q is odd and constructed from a Desarguesian plane \(\Sigma\) by a nest replacement of a \((q+1)\)- nest of reguli in \(\Sigma\), or (iii) q is odd and constructed from a Desarguesian plane \(\Sigma\) by a nest replacement of a \((q+1)\)-nest and the replacement of a set of mutually disjoint André nests in \(\Sigma\). Quite s…
Mathematics Subject Classification ID: 51E15
Mathematics Subject Classification ID: 51A40
Mathematics Subject Classification ID: 51J10
Mathematics Subject Classification ID: 51J20
zbMATH DE Number: 4194579
zbMATH Keywords: nests of reguli
zbMATH Keywords: translation planes
zbMATH Keywords: André plane
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1007/bf00147458
OpenAlex ID: W2017322494
description: scientific article; zbMATH DE number 4194579