Moufang polygons and irreducible spherical BN-pairs of rank 2. I
type: Item
instance of: scholarly article
title: Moufang polygons and irreducible spherical BN-pairs of rank 2. I
zbMATH Open document ID: 1030.20019
DOI: 10.1016/S0001-8708(02)00039-7
author: Katrin Tent
author: Hendrik Van Maldeghem
published in: Advances in Mathematics
publication date: 2003-05-04T00:00:00Z
review text: Let \(G\) be a group with an irreducible spherical BN-pair \((B,N)\) such that \(B=(B\cap N)U\) for a normal nilpotent subgroup \(U\) of \(B\). The authors show that \(G\) corresponds to a Moufang polygon \(\Gamma\) -- hence the action of \(G\) on \(\Gamma\) is known explicitly by the classification of \textit{J. Tits} and \textit{R. M. Weiss} [Moufang polygons, Springer (2002; Zbl 1010.20017)] -- under the assumption that \(\Gamma\) is not an octagon; this assumption on \(\Gamma\) is removed by \textit{K. Tent} [Adv. Math.181, No. 2, 308-320 (2004; Zbl 1044.20013)]. In view of the results of \textit{K. Tent} and \textit{H. Van Maldeghem} [Forum Math. 13, No. 6, 853-862 (2001; Zbl 0989.20026)], this means that one has now a classification of split BN-pairs of rank 2 which extends the well-known result of \textit{P. Fong} and \textit{G. M. Seitz} [Invent. Math. 21, 1-57 (1973; Zbl 0295.20048) and ibid. 24, 191-239 (1974; Zbl 0295.20049)] on BN-pairs in finite groups. The proof is achieved by elementary geometric arguments, in the style of \textit{R. Weiss} [Invent. Math. 51, 261-266 (1979; Zbl 0409.05033)].
Mathematics Subject Classification ID: 20E42
Mathematics Subject Classification ID: 05C25
Mathematics Subject Classification ID: 20F65
Mathematics Subject Classification ID: 51E24
zbMATH DE Number: 1903517
zbMATH Keywords: irreducible spherical BN-pairs
zbMATH Keywords: buildings
zbMATH Keywords: generalized polygons
zbMATH Keywords: Moufang polygons
zbMATH Keywords: split BN-pairs
reviewed by: Theo Grundhöfer
MaRDI profile type: Publication
cites work: Finite distance-transitive generalized polygons
cites work: The nonexistence of certain generalized polygons
cites work: Groups with a (B,N)-pair of rank 2. II
cites work: Finite groups with a split BN-pair of rank 1. I
cites work: Finite groups with a split BN-pair of rank 1. II
cites work: A geometric characterization of Moufang hexagons
cites work: Very Homogeneous Generalized n -Gons of Finite Morley Rank
cites work: Split BN-pairs of finite Morley rank
cites work: Split BN-pairs of rank 2: the octagons.
cites work: BN-pairs with projective or affine lines.
cites work: On irreducible \((B,N)\)-pairs of rank 2
cites work: Sur la trialité et certains groupes qui s'en déduisent
cites work: Buildings of spherical type and finite BN-pairs
cites work: Endliche Spiegelungsgruppen, die als Weylgruppen auftreten
cites work: Q4779974
cites work: Q4217275
cites work: Some consequences of a result of Brouwer
cites work: The nonexistence of certain Moufang polygons
description: scientific article; zbMATH DE number 1903517