A construction of the Held simple group
type: Item
instance of: scholarly article
title: A construction of the Held simple group
zbMATH Open document ID: 0986.20016
author: Gernot Stroth
published in: Journal of Algebra
publication date: 2002-02-26T00:00:00Z
review text: This article contains a new result within the larger frame of constructing sporadic simple groups by embedding an amalgam into some linear group, equivalently, a geometry into some vector space. Here, the sporadic simple group \(He\) discovered by Held is constructed as subgroup of \(\text{GL}(51,11)\). For this purpose, first matrices in \(\text{GL}(51,11)\) for the two maximal parabolics \(2^{1+6}L_3(2)\) and \(2^6.3S_6\) of the tilde geometry are constructed. Then, a subgroup \(X\cong S_4(4)\) is identified in the subgroup \(G\leq\text{GL}(51,11)\) generated by these matrices and a graph on the set of conjugates of \(X\) in \(G\) is defined. Considering the action of \(G\) on this graph, it is proved that \(|G|=|He|\). Finally, \(G\) is identified with \(He\) by proving that it satisfies the original definition of \(He\) by \textit{D. Held} [cf. J. Algebra 13, 253-296 (1969; Zbl 0182.04302)].
reviewed by: Corinna Wiedorn
Mathematics Subject Classification ID: 20D08
Mathematics Subject Classification ID: 20G40
zbMATH DE Number: 1565884
zbMATH Keywords: sporadic groups
zbMATH Keywords: geometries
zbMATH Keywords: amalgams
zbMATH Keywords: embeddings
zbMATH Keywords: Held simple group
MaRDI profile type: Publication
full work available at URL: jabr.2000.8554
OpenAlex ID: W1975306866
cites work: A computer-free construction of the third group of Janko
cites work: Graphs related to Held's simple group
cites work: Q3684278
cites work: A presentation and a representation of the Held group
cites work: On a parabolic system of type \(M_{24}\)
cites work: The simple groups related to \(M_{24}\)
cites work: Q4773867
cites work: A computer-free construction of \(J_4\)
cites work: A new finite simple group with abelian Sylow 2-subgroups and its characterization
cites work: Minimal 2-local geometries for the Held and Rudvalis sporadic groups
cites work: Minimal parabolic geometries for the sporadic groups
cites work: Q3369823
DOI: 10.1006/JABR.2000.8554
description: scientific article; zbMATH DE number 1565884