Generation of classical groups
type: Item
instance of: scholarly article
title: Generation of classical groups
zbMATH Open document ID: 0832.20029
DOI: 10.1007/BF01263536
author: Jan Saxl
author: Gunter Malle
author: Th. Weigel
published in: Geometriae Dedicata
publication date: 1996-02-22T00:00:00Z
review text: It is well known (modulo the classification of finite simple groups) that every finite group is generated by two elements and can not be generated by two involutions (= elements of order 2). The authors prove the following important Theorem A: every finite simple non-abelian group other than \(U_3(3)\) is generated by three involutions. \textit{A. Wagner} [Boll. Unione Mat. Ital., V. Ser., A 15, 431-439 (1978; Zbl 0401.20038)] has proved that \(U_3(3)\) can not be generated by any three of its involutions, but it is generated by four involutions. In fact, Theorem A has been proved previously for many series of simple groups; see the paper under review for references. Theorem B: every finite simple non- abelian group other than \(U_3(3)\) is generated by two elements, one of which is an involution and the other is a strongly real element (that is the product of two involutions).
Mathematics Subject Classification ID: 20D06
Mathematics Subject Classification ID: 20D05
Mathematics Subject Classification ID: 20F05
Mathematics Subject Classification ID: 20G40
zbMATH DE Number: 553940
zbMATH Keywords: classical groups
zbMATH Keywords: generators
zbMATH Keywords: classification of finite simple groups
zbMATH Keywords: generated by two elements
zbMATH Keywords: generated by three involutions
zbMATH Keywords: strongly real element
reviewed by: S. A. Syskin
MaRDI profile type: Publication
cites work: On the maximal subgroups of the finite classical groups
cites work: Involutions in Chevalley groups over fields of even order
cites work: Q5662096
cites work: Q3682663
cites work: The Symmetric Genus of Sporadic Groups
cites work: Q3684278
cites work: Q3472288
cites work: Generation of some orthogonal groups by a set of three involutions
cites work: Linear groups with large cyclic subgroups and translation planes
cites work: Centralizers of Semisimple Elements in Finite Twisted Groups of Lie Type
cites work: Q3251484
cites work: The lattices and Möbius functions of stable closed subrootsystems and hyperplane complements for classical Weyl groups
cites work: Transitive linear groups and linear groups which contain irreducible subgroups of prime order. II
cites work: The modular characters of the tits simple group and its automorphism group
cites work: Q3952291
cites work: Linear groups containing a Singer cycle
cites work: The probability of generating a finite classical group
cites work: The maximal subgroups of the finite 8-dimensional orthogonal groups \(P\Omega ^ +_ 8(q)\) and of their automorphism groups
cites work: Q3996618
cites work: On the minimal degrees of projective representations of the finite Chevalley groups
cites work: The characters of the finite unitary groups
cites work: Disconnected groups of Lie type as Galois groups.
cites work: Q3973544
cites work: Q4183492
cites work: Hereditary and semihereditary rings without nilpotent elements
cites work: Remarks on symbols, hooks and degrees of unipotent characters
cites work: Q4185037
cites work: Generation of exceptional groups of Lie-type
full work available at URL: https://doi.org/10.1007/bf01263536
OpenAlex ID: W2006879104
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description: scientific article; zbMATH DE number 553940