Overgroups of Cyclic Sylow Subgroups of Linear Groups

John Bamberg, Tim Penttila

Type: Article

Publication Date: 2008-06-04

Citations: 36

DOI: https://doi.org/10.1080/00927870802070108

Abstract

Abstract We use a theorem of Guralnick, Penttila, Praeger, and Saxl to classify the subgroups of the general linear group (of a finite dimensional vector space over a finite field) which are overgroups of a cyclic Sylow subgroup. In particular, our results provide the starting point for the classification of transitive m-systems; which include the transitive ovoids and spreads of finite polar spaces. We also use our results to prove a conjecture of Cameron and Liebler on irreducible collineation groups having equally many orbits on points and on lines. Key Words: Cameron–Liebler conjecture m-systemMatrix groupOvoidPrimitive prime divisorSpread2000 Mathematics Subject Classification: Primary 20G40Secondary 20C20, 20C33, 20C34 ACKNOWLEDGMENT We would like to thank Michael Giudici for many fruitful and stimulating conversations. This work forms part of an Australian Research Council Discovery Grant, for which the first author was supported. Notes Communicated by D. Easdown. Additional informationNotes on contributorsJohn Bamberg* *Current affiliation: Department of Pure Mathematics and Computer Algebra, Ghent University, Krijgslaan 281-S22, 9000 Ghent, Belgium. Tim Penttila** **Current affiliation: Department of Mathematics, Colorado State University, Fort Collins, CO 80523, USA; E-mail: [email protected]

Locations

CiteSeer X (The Pennsylvania State University) - View - PDF
Communications in Algebra - View

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+	A Hauptsatz of L. E. Dickson and Artin-Schreier extensions.	1980	Robert C. Valentini Manohar L. Madan
+	The Subgroup Structure of the Finite Classical Groups	1990	Peter B. Kleidman Martin W. Liebeck
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+	The maximal subgroups of the chevalley groups G2(q) with q odd, the Ree groups 2G2(q), and their automorphism groups	1988	Peter B. Kleidman
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