A simple construction of the Ree groups of type \(^2F_4\).
type: Item
instance of: scholarly article
title: A simple construction of the Ree groups of type \(^2F_4\).
zbMATH Open document ID: 1206.20015
published in: Journal of Algebra
publication date: 2010-04-21T00:00:00Z
review text: The large Ree groups \(^2F_4(2^{2n+1})\) were originally discovered by \textit{R. Ree} [Am. J. Math. 83, 432-462 (1961; Zbl 0104.24705)] by twisting the groups \(F_4(2^{2n+1})\) of Lie type with an outer automorphism. A construction in terms of buildings was given by \textit{J. Tits} [Am. J. Math. 105, 539-594 (1983; Zbl 0521.20016)], and another in terms of Moufang polygons by \textit{J. Tits} and \textit{R. Weiss} in Chapter 32 of ``Moufang Polygons'' [Springer Monographs in Mathematics. Berlin: Springer (2002; Zbl 1010.20017)]. In the present paper, the author gives an elementary construction of the groups \(^2F_4(q)\), avoiding the need for the machinery of Lie algebras, algebraic groups, or buildings. This construction was inspired by author's paper [Proc. Edinb. Math. Soc.., II. Ser. 53, No. 2, 531-542 (2010; Zbl 1205.20012)], in which he gave a new construction of the Ree groups of type \(^2G_2\). The author gives an explicit description of the generalized octagon whose automorphism group is \(^2F_4(q)\). On the way, he obtains generators for many of the maximal subgroups. The author remarks that \textit{K. Coolsaet} [in Innov. Incidence Geom. 1, 67-131 (2005; Zbl 1102.51002), Adv. Geom. 7, No. 3, 423-452 (2007; Zbl 1176.51002) and Des. Codes Cryptography 47, No. 1-3, 75-97 (2007; Zbl 1186.51004)] obtained independently a description of the Ree-Tits octagon which is almost identical to his description.
reviewed by: Anatoli Kondrat'ev
Mathematics Subject Classification ID: 20D06
Mathematics Subject Classification ID: 20G40
Mathematics Subject Classification ID: 51E12
zbMATH DE Number: 5696917
zbMATH Keywords: groups of Lie type
zbMATH Keywords: finite simple groups
zbMATH Keywords: Ree groups
zbMATH Keywords: generalized octagons
author: Robert A. Wilson
MaRDI profile type: Publication
full work available at URL: j.jalgebra.2009.11.015
OpenAlex ID: W2038636116
cites work: Q5662096
cites work: Algebraic structure of the perfect Ree-Tits generalized octagon
cites work: On a 25-dimensional embedding of the Ree–Tits generalized octagon
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cites work: Matrix generators for exceptional groups of Lie type
cites work: The maximal subgroups of \({}^ 2F_ 4(q^ 2)\)
cites work: Constructions for octonion and exceptional Jordan algebras
cites work: Q4217275
cites work: A family of simple groups associated with the simple lie algebra of type (𝐹₄)
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cites work: Q4779974
cites work: A new construction of the Ree groups of type 2G2
cites work: A new approach to the Suzuki groups
DOI: 10.1016/J.JALGEBRA.2009.11.015
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description: scientific article; zbMATH DE number 5696917