Products of conjugacy classes in Chevalley groups. I: Extended covering numbers
type: Item
instance of: scholarly article
title: Products of conjugacy classes in Chevalley groups. I: Extended covering numbers
zbMATH Open document ID: 1007.20041
DOI: 10.1007/BF02764077
author: N. L. Gordeev
author: Jan Saxl
published in: Israel Journal of Mathematics
publication date: 2002-10-31T00:00:00Z
review text: Let \(G\) be a group. The `covering number' \(\text{cn}(G)\) is the smallest integer \(m\) such that \(C^m=G\) for every conjugacy class \(C\) of \(G\) which is not contained in any proper normal subgroup of \(G\). The `extended covering number' \(\text{ecn}(G)\) is the smallest integer \(e\) such that the product \(C_1C_2\cdots C_e=G\), whenever \(C_1,C_2,\dots,C_e\) are conjugacy classes of \(G\) not contained in any proper normal subgroup of \(G\). The main result obtained by the authors is the following theorem: There is a constant \(e\) such that for any Chevalley group \(G=G(F)\) (over any field \(F\)), the inequality \(\text{ecn}(G)\leq e\cdot\text{rank}(G)\) holds. This generalizes an earlier result; namely, there is a constant \(c\) such that \(\text{cn}(G)\leq c\cdot\text{rank}(G)\) for every quasi-simple Chevalley group \(G\) [\textit{E. W. Ellers, N. Gordeev, M. Herzog}, Isr. J. Math. 111, 339-372 (1999; Zbl 0941.20050)]. The constant \(e\), as constructed in the proof, is fairly large in general. Better estimates are obtained in special cases, e.g. for \(\text{SL}_n(F)\). The authors also show: Let \(G\) be a finite crystallographic Chevalley group (untwisted or twisted). Then every noncentral conjugacy class of \(G\) has a nonempty intersection with a general Coxeter cell of the Bruhat decomposition of \(G\).
reviewed by: Erich W. Ellers
Mathematics Subject Classification ID: 20G15
Mathematics Subject Classification ID: 20F05
Mathematics Subject Classification ID: 20E45
zbMATH DE Number: 1799793
zbMATH Keywords: extended covering numbers
zbMATH Keywords: Chevalley groups
zbMATH Keywords: conjugacy classes
zbMATH Keywords: Coxeter cells
zbMATH Keywords: Bruhat decompositions
MaRDI profile type: Publication
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full work available at URL: https://doi.org/10.1007/bf02764077
OpenAlex ID: W2051695580
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description: scientific article; zbMATH DE number 1799793