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Eigenspaces of Linear Collineations

Eigenspaces of Linear Collineations Adv. Geom. 5 (2005), 71­79 ( de Gruyter 2005 Chat Yin Ho* (Communicated by the Managing Editors) Dedicated to Professor William M. Kantor 1 Introduction The Fundamental theorem of projective geometry indicates the importance of projective planes. Among projective planes, translation planes have the richest algebraic structure. A long standing question in the study of translation planes is the following: Which non-abelian simple groups can be collineation groups for a finite translation plane of odd order (see, for example [11]). A finite translation plane is a vector space over a field endowed with a spread. In this paper, we classify non-abelian simple collineation groups in the case in which the cardinality of the field is odd and congruent to 1 modulo 3 (Theorem A, below), and we improve a result in [8] (Theorem B, below). Theorem A. A non-abelian simple collineation group in the translation complement of a finite translation plane over a field with odd characteristic and cardinality congruent to 1 modulo 3 must be isomorphic to one of the following simple groups: L 2 ð2 n Þ; Szð2 n Þ; U3 ð2 n Þ, n odd. Theorem B. Let G be a non-abelian simple collineation group http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Advances in Geometry de Gruyter

Eigenspaces of Linear Collineations

Advances in Geometry , Volume 5 (1) – Jan 1, 2005

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Publisher
de Gruyter
Copyright
Copyright © 2005 by the
ISSN
1615-715X
eISSN
1615-7168
DOI
10.1515/advg.2005.5.1.71
Publisher site
See Article on Publisher Site

Abstract

Adv. Geom. 5 (2005), 71­79 ( de Gruyter 2005 Chat Yin Ho* (Communicated by the Managing Editors) Dedicated to Professor William M. Kantor 1 Introduction The Fundamental theorem of projective geometry indicates the importance of projective planes. Among projective planes, translation planes have the richest algebraic structure. A long standing question in the study of translation planes is the following: Which non-abelian simple groups can be collineation groups for a finite translation plane of odd order (see, for example [11]). A finite translation plane is a vector space over a field endowed with a spread. In this paper, we classify non-abelian simple collineation groups in the case in which the cardinality of the field is odd and congruent to 1 modulo 3 (Theorem A, below), and we improve a result in [8] (Theorem B, below). Theorem A. A non-abelian simple collineation group in the translation complement of a finite translation plane over a field with odd characteristic and cardinality congruent to 1 modulo 3 must be isomorphic to one of the following simple groups: L 2 ð2 n Þ; Szð2 n Þ; U3 ð2 n Þ, n odd. Theorem B. Let G be a non-abelian simple collineation group

Journal

Advances in Geometryde Gruyter

Published: Jan 1, 2005

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