On flag collineations of finite projective planes
type: Item
instance of: scholarly article
title: On flag collineations of finite projective planes
zbMATH Open document ID: 0613.51005
DOI: 10.1007/BF01221940
author: Adilson Goncalves
author: Chat Yin Ho
published in: Journal of Geometry
publication date: 1987-01-01T00:00:00Z
review text: A flag collineation is a collineation which leaves some incident point- line pair invariant. The main result is the following: Let \(\Pi\) be a finite projective plane of order n and let G be a collineation group of \(\Pi\) which contains a non-trivial perspectivity. Suppose that G has a subgroup H of order n such that for some incident point-line pair P, \(\ell\) every element of H is a flag collineation leaving P, \(\ell\) invariant and that H has order n. Then one of the following holds: (1) G has an invariant point-line pair. (2) G has an invariant line but no invariant points and is a translation plane. (3) The dual of (2). (4) The plane \(\Pi\) is Desarguesian. A group G of collineations of a projective plane is said to be strongly irreducible if there is no invariant point, line, triangle, or proper subplane. The arguments depend on known results about strongly irreducible groups which contain non-trivial perspectivities (central collineations).
Mathematics Subject Classification ID: 51A35
Mathematics Subject Classification ID: 20B25
zbMATH DE Number: 3992299
zbMATH Keywords: flag collineations of finite projective planes
reviewed by: Theodore G. Ostrom
MaRDI profile type: Publication
cites work: On the structure of finite collineation groups of projective planes
cites work: Perspectivities in irreducible collineation groups of projective planes. I
cites work: Perspectivities in irreducible collineation groups of projective planes, II
cites work: On unitary collineation groups
cites work: On the Structure of Collineation Groups of Finite Projective Planes
cites work: Linear transformations and collineations of translation planes
cites work: Endliche zyklische Ebenen
cites work: On finite simple groups containing perspectivities
cites work: On Chevalley-groups acting on projective planes
cites work: On perspectivities of finite projective planes
cites work: A theorem on doubly transitive permutation groups
full work available at URL: https://doi.org/10.1007/bf01221940
OpenAlex ID: W1966220754
description: scientific article; zbMATH DE number 3992299