Collineation groups with perspectivities.
type: Item
instance of: scholarly article
title: Collineation groups with perspectivities.
zbMATH Open document ID: 1074.51001
DOI: 10.1016/S0021-8693(03)00492-7
published in: Journal of Algebra
publication date: 2004-05-27T00:00:00Z
review text: Let \({\mathcal T}\) be a finite translation plane. The collineation group of \({\mathcal T}\) is a semi-direct product of the translation group and the translation complement. The translation complement is a semi-linear transformation group. A subgroup of the translation complement consisting of linear collineations is called a linear collineation group of \({\mathcal T}\). In the paper under review Ho provides some remarkable results about linear collineation groups of finite translation planes. His main results read as follows. Theorem 1. Let \(G\) be a simple linear collineation group containing a perspectivity of a finite translation plane. (i) If \(G\) is isomorphic to an alternating group \(A_n\), then \(n=5\), and the translation plane is of even order. (ii) \(G\) is not isomorphic to any one of the 26 sporadic finite simple groups. Theorem 2. Let \(G\) be a simple linear collineation group containing an elation of a finite translation plane. (i) The translation plane is of even order and \(G \cong L_2(u)\) or \(G \cong Sz(u)\), where \(u\) is a power of 2. (ii) If all involutions of \(G\) are Baer involutions, then any perspectivity in \(G\) has an order dividing \(u + 1\).
reviewed by: Johannes Ueberberg
Mathematics Subject Classification ID: 51A40
Mathematics Subject Classification ID: 51E15
zbMATH DE Number: 2066832
zbMATH Keywords: translation planes
zbMATH Keywords: linear collineation groups
author: Chat-Yin Ho
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 2066832