Fixed point ratios in actions of finite exceptional groups of Lie type.
type: Item
instance of: scholarly article
title: Fixed point ratios in actions of finite exceptional groups of Lie type.
zbMATH Open document ID: 1058.20001
author: Ross Lawther
author: Martin W. Liebeck
author: Gary M. Seitz
published in: Pacific Journal of Mathematics
publication date: 2003-02-13T00:00:00Z
review text: Let \(G\) be a finite group acting transitively on a set \(\Omega\) and \(x\in\Omega\). The number \[ \text{fpr}(x,\Omega)=\tfrac{\text{fix}_\Omega(x)}{|\Omega| }, \] where \(\text{fix}_\Omega(x)\) is the number of fixed points of \(x\) on \(\Omega\), is called the fixed point ratio of \(x\). Fixed point ratios have been much studied in recent years, and applied to a number of different problems, particularly in the case where \(G\) almost simple. In papers of \textit{D. Gluck} and \textit{K. Magaard} [Proc. Lond. Math. Soc., III. Ser. 71, No. 3, 547-584 (1995; Zbl 0865.20012)], \textit{M. W. Liebeck} and \textit{A. Shalev} [J. Am. Math. Soc. 12, No. 2, 497-520 (1999; Zbl 0916.20003)], \textit{D. Frohardt} and \textit{K. Magaard} [Geom. Dedicata 82, No. 1-3, 21-104 (2000; Zbl 0970.20003)] and \textit{R. M. Guralnick} and \textit{W. M. Kantor} [J. Algebra 234, No. 2, 743-792 (2000; Zbl 0973.20012)], upper bounds of fixed point ratios in classical groups \(G\) were obtained and applied to various problems. \textit{M. W. Liebeck} and \textit{J. Saxl} [Proc. Lond. Math. Soc., III. Ser. 63, No. 2, 266-314 (1991; Zbl 0696.20004)] obtained such a general upper bound of \(4/3q\) for groups of Lie type over \(F_q\) (with a few exceptions). These bounds were used by \textit{M. W. Liebeck} and \textit{A. Shalev} [loc. cit.] and by \textit{D. Frohardt} and \textit{K. Magaard} [Ann. Math. (2) 154, No. 2, 327-345 (2001; Zbl 1004.20001)] to prove the Guralnick-Thompson monodromy group conjecture. \textit{D. Frohardt} and \textit{K. Magaard} [Commun. Algebra 30, No. 2, 571-602 (2002; Zbl 1019.20011)] obtained better than \(4/3q\) bounds for exceptional groups of Lie type of rank at most 2. In the present paper, the authors obtain stronger upper bounds for fixed point ratios of every group \(G\) with the socle \(L\) isomorphic to a finite simple exceptional group of Lie type. The proof of the main result is divided into several cases, giving upper bounds for \(\text{fpr}(x,\Omega)\)…
Mathematics Subject Classification ID: 20B15
Mathematics Subject Classification ID: 20G40
Mathematics Subject Classification ID: 20D06
Mathematics Subject Classification ID: 20C33
Mathematics Subject Classification ID: 20D60
zbMATH DE Number: 1868167
zbMATH Keywords: finite exceptional groups of Lie type
zbMATH Keywords: fixed point ratios
zbMATH Keywords: transitive actions
zbMATH Keywords: primitive actions
reviewed by: Anatoli Kondrat'ev
MaRDI profile type: Publication
OpenAlex ID: W2015901827
Wikidata QID: Q115230662
DOI: 10.2140/PJM.2002.205.393
description: scientific article; zbMATH DE number 1868167