Estimating proportions of elements in finite groups of Lie type.
type: Item
instance of: scholarly article
title: Estimating proportions of elements in finite groups of Lie type.
zbMATH Open document ID: 1223.20046
author: Alice C. Niemeyer
author: Cheryl E. Praeger
published in: Journal of Algebra
publication date: 2010-07-20T00:00:00Z
review text: The authors formalize the notion of a subset of a finite group of Lie type possessing certain closure properties and prove that certain closure properties on subsets \(Q\) of a finite group \(G\) of Lie type enable the ratio \(\frac{|Q|}{|G|}\) to be determined by finding the proportions of elements of \(Q\) in the maximal tori of \(G\) and the proportions of certain related subsets of the Weyl group. Moreover, they prove fundamental results about these subsets \(Q\), including those necessary for moving between these groups and their automorphism groups, normal subgroups and central quotients. As example of application of these new methods, the authors derive upper and lower bounds for the proportion of elements with order divisible by certain natural sets of primes, for the finite classical groups, thereby improving existing results. Precisely, in Section 5, the authors show the power of their main theorem (Thm. 1.3) by rederiving and extending the results of \textit{A. C. Niemeyer} and \textit{C. E. Praeger} [Proc. Lond. Math. Soc., III. Ser. 77, No. 1, 117-169 (1998; Zbl 0893.20035)] on the proportion of primitive prime divisor elements in classical groups.
reviewed by: Fiorenza Morini
Mathematics Subject Classification ID: 20G40
Mathematics Subject Classification ID: 20D06
Mathematics Subject Classification ID: 20D60
Mathematics Subject Classification ID: 20F05
Mathematics Subject Classification ID: 11E57
zbMATH DE Number: 5758111
zbMATH Keywords: estimating proportions of elements
zbMATH Keywords: primitive prime divisor elements
zbMATH Keywords: finite groups of Lie type
zbMATH Keywords: finite classical groups
MaRDI profile type: Publication
full work available at URL: j.jalgebra.2010.03.018
OpenAlex ID: W2083225892
Wikidata QID: Q56987668
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DOI: 10.1016/J.JALGEBRA.2010.03.018
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description: scientific article; zbMATH DE number 5758111