Global Cayley maps and conjugacy class sizes of maximal unipotent subgroups of finite simple groups of Lie type.
type: Item
instance of: scholarly article
title: Global Cayley maps and conjugacy class sizes of maximal unipotent subgroups of finite simple groups of Lie type.
zbMATH Open document ID: 1241.20021
published in: Journal of Pure and Applied Algebra
publication date: 2012-01-24T00:00:00Z
review text: Let \(p\) be a prime and \(G\) be a simple affine algebraic group defined over an algebraic closure \(\overline{\mathbb F}_p\) of the finite field \(\mathbb F_p\). We may assume that \(G\) is a closed subgroup of \(\text{GL}_d(\overline{\mathbb F}_p)\) for some \(d\). Given a power \(q_0\) of \(p\), define \(\sigma_{q_0}(x_{ij})=(x_{ij}^{q_0})\). Then \(\sigma_{q_0}\) is an endomorphism of \(\text{GL}_d(\overline{\mathbb F}_p)\). An endomorphism \(F\) of \(G\) is called a Frobenius endomorphism if there exists an integer \(m\) such that \(F^m(x)=\sigma_{q_0}(x)\) for all \(x\in G\). Define \(q=q_0^{1/m}\) as the level of \(F\). Let \(G_F\) be the group of elements of \(G\) fixed by \(F\). \(G_F\) is as a rule a finite simple group of Lie type. Let \(U_F\) be a \(p\)-Sylow (maximal unipotent) subgroup of \(G_F\). As is well-known, \(|U_F|\) is a power of \(q\) when \(p\) is large enough. The first author [Commun. Algebra 27, No. 5, 2511-2519 (1999; Zbl 0923.20040)] proved that any good prime is ``large enough'' when \(G\) is a classical group or of type \(G_2\). This led to the conjecture that \(|U_F|\) is a power of the level \(q\) of \(F\) when \(p\) is good without further restrictions on \(G\). In the present paper, the authors prove this conjecture. If \(p\) is a bad prime for \(G\), then the conjecture will be false in general. In the last part of the paper, the authors investigate the existence of power series inducing isomorphisms (global Cayley maps) between \(U_F\) and its Lie algebra.
reviewed by: Anatoli Kondrat'ev
Mathematics Subject Classification ID: 20D06
Mathematics Subject Classification ID: 20E45
Mathematics Subject Classification ID: 20D20
Mathematics Subject Classification ID: 20G40
zbMATH DE Number: 5999962
zbMATH Keywords: finite simple groups of Lie type
zbMATH Keywords: maximal unipotent subgroups
zbMATH Keywords: conjugacy class sizes
zbMATH Keywords: global Cayley maps
Wikidata QID: Q115345524
author: Andrea Previtali
author: Th. Weigel
describes a project that uses: Magma
MaRDI profile type: Publication
full work available at URL: j.jpaa.2011.06.005
OpenAlex ID: W2042482479
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DOI: 10.1016/J.JPAA.2011.06.005
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description: scientific article; zbMATH DE number 5999962