Transformation of designs and other incidence structures
type: Item
instance of: scholarly article
title: Transformation of designs and other incidence structures
zbMATH Open document ID: 0773.51007
DOI: 10.1007/BF00182951
author: Pasquale Quattrocchi
author: Luigi Antonio Rosati
published in: Geometriae Dedicata
publication date: 1993-04-01T00:00:00Z
review text: In the 1960's there was a great need for different examples of projective and affine planes. The reviewer [Trans. Am. Math. Soc. 111, 1-18 (1964; Zbl 0117.373)] discovered a process, called derivation, which yielded new planes from known planes of finite square order. Later, the reviewer generalized this process to one, called net replacement, which did not require square order. [See the reviewer, ``Finite Translation Planes'' (1970; Zbl 0205.499).](Although the reviewer describes net replacement for translation planes, it works for arbitrary planes in exactly the same way.) Recently, \textit{E. F. Assmus} jun. and \textit{J. D. Key} [J. Geom. 37. No. 1/2, 3-16 (1990; Zbl 0705.51011)] described another procedure for generating new planes from known (translation) planes. The authors of the present article describe a procedure which generalizes the reviewer's replacement method to incidence structures. Although the authors' examples are known examples of net replacement, their procedure is a natural extension of the reviewer's one and should give rise to some interesting incidence structures. The authors apply their procedure to show that all known sharply 3-transitive finite permutation sets containing the identity can be obtained from the group \(PGL(2,q)\).
Mathematics Subject Classification ID: 51E30
Mathematics Subject Classification ID: 05B20
zbMATH DE Number: 147299
zbMATH Keywords: derivation
zbMATH Keywords: net replacement
reviewed by: Theodore G. Ostrom
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1007/bf00182951
OpenAlex ID: W1977614368
description: scientific article; zbMATH DE number 147299