A family of translation planes of order \(q^{2m+1}\) with two orbits of length 2 and \(q^{2m+1}-1\) on \(l_ \infty\)
type: Item
instance of: scholarly article
title: A family of translation planes of order \(q^{2m+1}\) with two orbits of length 2 and \(q^{2m+1}-1\) on \(l_ \infty\)
zbMATH Open document ID: 0765.51001
DOI: 10.1007/BF00147547
author: Chihiro Suetake
published in: Geometriae Dedicata
publication date: 1992-09-27T00:00:00Z
review text: The author constructs a class of translation planes of order \(q^{2m+1}\) where \(q\) is an odd prime power and \(m\geq 1\). These translation planes \(\pi\) have the following properties: \(\pi\) has order \(2m+1\) over its kernel. If \(q^{2m+1}\neq 3^ 3\) then the linear translation complement has two orbits on \(\ell_ \infty\): one of length 2 and the other of length \(q^{2m+1}-1\). Many of these planes are not generalized André planes. If \(m=1\) then the above class of translation planes is contained in the class of planes constructed by Hiramine in 1986. If \(q^{2m+1}=3^ 3\) then each plane of this class is isomorphic to the Hering plane of order 27.
reviewed by: Helmut Mäurer
Mathematics Subject Classification ID: 51A40
zbMATH DE Number: 57581
zbMATH Keywords: translation planes
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 57581