On a class of translation planes of square order
type: Item
instance of: scholarly article
title: On a class of translation planes of square order
zbMATH Open document ID: 0621.51007
DOI: 10.1016/0012-365X(87)90130-0
author: M. L. Narayana Rao
author: K. Satyanarayana
author: G. Vithal Rao
published in: Discrete Mathematics
publication date: 1987-01-01T00:00:00Z
review text: \textit{V. Jha} and the reviewer [Can. J. Math. 33, 1487-1497 (1981; Zbl 0474.51004)] considered a translation plane \(\pi\) of order \(q^ d\) with a collineation group G fixing the origin and leaving a set \(\Delta\) of \(q+1\) components invariant while acting transitively on the remaining \(q^ d-q\) components. In that article and additional articles by Jha and others the plane \(\pi\) and the group G is determined under additional conditions. [See \textit{V. Jha}, Finite geometries, Lect. Notes Pure Appl. Math. 82, 237-242 (1983; Zbl 0524.51005).] In almost all cases it was proven that \(\pi\) is Hall or Desarguesian. The authors of the present article under review give a class of such planes in which each plane has order \(q^ 2\) and the translation complement C fixes one component and divides the remaining components into two orbits of length q and \(q^ 2- q\) respectively. Furthermore, the group G is completely determined. Although the authors show that these planes are not the planes discussed by \textit{V. Jha} [Arch. Math. 37, 377-384 (1981; Zbl 0455.51003)] and \textit{S. D. Cohen} and \textit{M. J. Ganley} [Q. J. Math., Oxf. II. Ser. 35, 101- 113 (1984; Zbl 0541.51005)] in connection with the problem discussed above, it is not clear that they are new. It is possible that they are associated with known planes by a standard procedure like derivability or homology replacement. Even if this were true, still it is new that they fulfill the condition Jha calls \(\Delta\)-transitive.
Mathematics Subject Classification ID: 51A40
Mathematics Subject Classification ID: 51E15
zbMATH DE Number: 4007105
zbMATH Keywords: translation plane
zbMATH Keywords: translation complement
MaRDI profile type: Publication
cites work: Linear representations of projective planes in projective spaces
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full work available at URL: 0012-365x(87)90130-0
OpenAlex ID: W1976672146
description: scientific article; zbMATH DE number 4007105