A class of d-dimensional flag transitive translation planes
type: Item
instance of: scholarly article
title: A class of d-dimensional flag transitive translation planes
zbMATH Open document ID: 0666.51003
DOI: 10.1007/BF01224239
author: M. L. Narayana Rao
author: K. Kuppu Swamy Rao
author: K. V. Durga Prasad
published in: Journal of Geometry
publication date: 1989-01-01T00:00:00Z
review text: The curious title of this paper does not refer to planes of dimension larger than 2... In fact the authors study several cases of the classical construction of (affine) translation planes by means of spreads [see for instance \textit{D. R. Hughes} and \textit{F. C. Piper}, Projective Planes (1973; Zbl 0267.50018), Ch. VII]; each component of the spread is a d- dimensional subspace of a 2d-dimensional vector space over GF(q), where \(q=p^ r\) is a prime power. In this paper, d is an odd positive integer such that \((q-1,d)=1;\) under certain conditions imposed to the spread, it yields a plane which has a subgroup of collineations acting transitively on the set of flags (pairs of incident point line), and which is in certain cases desarguesian, in other cases non-desarguesian. The planes of order 27 and 125 previously found by the Raos are particular cases of those constructed in this paper. Reviewer's comments: (i) The reader will easily correct a few misprints (re-establishing ``non-desarguesian'' in the first line, ``Let \(\alpha\) '' in 1.12 of p. 140, the symbol for intersection in 1.3 of p. 141), but others may be hidden. (ii) The proofs are difficult to follow particularly because some notations (``M(V)'', ``D(V)'', ``E(V)'') are not defined while not being of general use in the field. (iii) It is nowhere said that the paper is about affine planes...
Mathematics Subject Classification ID: 51A40
Mathematics Subject Classification ID: 51E15
Mathematics Subject Classification ID: 05B25
Mathematics Subject Classification ID: 20B25
zbMATH DE Number: 4090258
zbMATH Keywords: flag transitive
zbMATH Keywords: spreads
zbMATH Keywords: non-desarguesian
zbMATH Keywords: affine planes
reviewed by: Jean Porte
MaRDI profile type: Publication
cites work: Eine nicht-desarguesche zweifach transitive affine Ebene der Ordnung 27
cites work: A New Flag Transitive Affine Plane of Order 27
cites work: Q3884734
cites work: Q3317664
full work available at URL: https://doi.org/10.1007/bf01224239
OpenAlex ID: W2055186477
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description: scientific article; zbMATH DE number 4090258