On translation planes of order \(q^ 3\) that admit a collineation group of order \(q^ 3\)
type: Item
instance of: scholarly article
title: On translation planes of order \(q^ 3\) that admit a collineation group of order \(q^ 3\)
zbMATH Open document ID: 0843.51005
DOI: 10.1007/BF02179084
author: Kenzi Akiyama
author: Chihiro Suetake
published in: Geometriae Dedicata
publication date: 1995-01-01T00:00:00Z
review text: The authors study translation planes of order \(k\), that admit a collineation group \(G\) of order \(k\) in the linear translation complement, for \(k = q^3\) (\(q\) -- a prime power). With the following additional assumptions all such planes are determined. Authors require that the kernel of a plane contains \(GF(q)\), a group \(G\) fixes one point at infinity and acts transitively on the remaining points at infinity, the elation group \(E\) of \(G\) is of order \(q\), and \(G/E\), is abelian. Authors show a general construction of spread sets which yields exactly all the planes in question. The definitions of spread make use of certain parameters of \(\text{GF} (q)\), so the construction is not really effective; to obtain a desired plane one must find concrete parameters which really yield a spread set. On the other hand, a matrix representation of the corresponding groups \(G\) and \(E\) is given. Next, the authors consider the special case of prime \(q\). Then \(E\) is of order \(q^3\) or the requirements on \(E\) follow from the remaining assumptions about \(G\), and a plane in question is either a semifield plane (of order \(q^3\)), or the Sherk plane of order 27.
reviewed by: K. Prażmowski
Mathematics Subject Classification ID: 51A40
Mathematics Subject Classification ID: 51E15
zbMATH DE Number: 806218
zbMATH Keywords: translation planes
zbMATH Keywords: collineation group
zbMATH Keywords: spread set
MaRDI profile type: Publication
cites work: On the Sherk plane
cites work: A structure theory for two-dimensional translation planes of order \(q^2\) that admit collineation groups of order \(q^2\)
cites work: A characterization of the generalized twisted field planes
cites work: Translation planes of order 27
cites work: A characterization of ''likeable'' translation planes
cites work: A note on finite semifield planes that admit affine homologies
cites work: Translation planes of order \(q^ 2\) that admit a collineation group of order \(q^ 2\)
cites work: On point-transitive affine planes
cites work: Q3216750
cites work: Q3891315
cites work: On a Kaplansky conjecture concerning three-dimensional division algebras over a finite field
cites work: On Sherk's plane of order 27
cites work: On quasifields
cites work: Polynomials and linear transformations over finite fields.
full work available at URL: https://doi.org/10.1007/bf02179084
OpenAlex ID: W4245694126
description: scientific article; zbMATH DE number 806218