Minimal parabolic geometries for the sporadic groups
type: Item
instance of: scholarly article
title: Minimal parabolic geometries for the sporadic groups
zbMATH Open document ID: 0574.20012
DOI: 10.1016/S0195-6698(84)80020-7
published in: European Journal of Combinatorics
publication date: 1984-01-01T00:00:00Z
review text: Most of the finite simple groups are of Lie type, and these act on geometries called Tits buildings [\textit{J. Tits}, Buildings of spherical type, Lect. Notes Math. 386 (1974; Zbl 0295.20047)]. Analogous geometries for sporadic simple groups were first studied by \textit{F. Buekenhout} [J. Comb. Theory, Ser. A 27, 121-151 (1979; Zbl 0419.51003)] with further examples given by \textit{W. Kantor} [Eur. J. Comb. 2, 239-247 (1981; Zbl 0514.51006)]. The idea of concentrating on geometries defined by p-local subgroups was introduced by \textit{M. Ronan} and the reviewer [Proc. Symp. Pure Math. 37, 283-289 (1980; Zbl 0478.20015)]. Whereas that paper considered ''maximal parabolics'' as basic, the present paper begins with ''minimal parabolics'', subgroups minimal with respect to containing some fixed subgroup B (typically the normalizer of a Sylow p-group). A set of such subgroups defines a chamber system in the sense of \textit{J. Tits} [in The Geometric Vein, 519-547 (1982; Zbl 0496.51001)]. The authors determine, for the various sporadic groups and their prime divisors, such minimal parabolic geometries. This work has been very influential in the modern study of groups and geometries. It contains much useful information on the subgroup structure of sporadic groups.
Mathematics Subject Classification ID: 20D08
Mathematics Subject Classification ID: 20F65
Mathematics Subject Classification ID: 51A05
Mathematics Subject Classification ID: 20D30
zbMATH DE Number: 3916544
zbMATH Keywords: finite simple groups
zbMATH Keywords: geometries
zbMATH Keywords: Tits buildings
zbMATH Keywords: sporadic simple groups
zbMATH Keywords: p-local subgroups
zbMATH Keywords: chamber system
zbMATH Keywords: parabolic geometries
author: Mark A. Ronan
author: Gernot Stroth
MaRDI profile type: Publication
cites work: Tits geometries over gf(2) Defined by groups over gf(3)
cites work: Diagrams for geometries and groups
cites work: Q3668381
cites work: Locally polar spaces and related rank 3 groups
cites work: The maximal subgroups of the sporadic simple group of Held
cites work: Q3875938
cites work: Automorphisms of trivalent graphs
cites work: The maximal subgroups of Conway's group C\(_3\) and McLaughlin's group
cites work: Maximal subgroups of the Hall-Janko-Wales group
cites work: The maximal subgroups of Janko's simple group of order 50, 232, 960
cites work: Q4128974
cites work: Some geometries that are almost buildings
cites work: Evidence for a new finite simple group
cites work: The subgroup structure of the Higman-Sims simple group
cites work: Minimal 2-local geometries for the Held and Rudvalis sporadic groups
cites work: Some sporadic geometries related to PG(3,2)
cites work: Some Evidence for the Existence of a New Simple Group
cites work: On Thompson's simple group
cites work: Locally truncated buildings and M//(24)
cites work: Q3920241
cites work: Q3934592
cites work: Near \(n\)-gons and line systems
cites work: A characterization of Fischer's sporadic simple group of the order \(2^{41}\cdot 3^{13}\cdot 5^2\cdot 7\cdot 11\cdot 13\cdot 17\cdot 19\cdot 23\cdot 31\cdot 47\)
cites work: Buildings of spherical type and finite BN-pairs
cites work: Q3961400
cites work: Edge-symmetric graphs
cites work: The complex Leech lattice and maximal subgroups of the Suzuki group
cites work: The Geometry and Maximal Subgroups of the Simple Groups of A. Rudvalis and J. Tits
cites work: The maximal subgroups of Conway's group. 2
cites work: The maximal subgroups of Conway's group \(Co_ 1\)
cites work: On maximal subgroups of the Fischer group Fi22
description: scientific article; zbMATH DE number 3916544