Non-abelian representations of some sporadic geometries
type: Item
instance of: scholarly article
title: Non-abelian representations of some sporadic geometries
zbMATH Open document ID: 0853.51008
author: Alexander A. Ivanov
author: Dmitrii V. Pasechnik
author: S. Shpectorov
published in: Journal of Algebra
publication date: 1996-07-09T00:00:00Z
review text: For a point-line geometry \(G = (P,L)\) with exactly three points per line, (the non-abelian) universal representation group is defined by \(R(G) = \langle z_p, p \in P \mid z^2_p = 1\) for \(p \in P\), \(z_p z_q z_r = 1\) for \(\{p, q, r\} \in L\rangle\). The commutator factor group \(V\) of \(R(G)\) is a vector space over \(GF(2)\), giving a nice module for \(G\), provided that \(V\) is not trivial. The authors prove the following main result: if \(G\) is the 2-local parabolic geometry of the sporadic simple group \(F_1\) (the monster) or of \(F_2\) (the baby monster), then \(R(G) \cong F_2\) or \(2 \cdot F_2\), respectively. The authors also determine the universal representation groups of some geometries related to the Mathieu groups \(M_{22}\), \(M_{24}\) and to the Conway groups \(Co_1\) and \(Co_2\).
Mathematics Subject Classification ID: 51E24
Mathematics Subject Classification ID: 20D08
Mathematics Subject Classification ID: 20C34
zbMATH DE Number: 883772
zbMATH Keywords: universal representation group
zbMATH Keywords: sporadic group
zbMATH Keywords: monster
zbMATH Keywords: baby monster
zbMATH Keywords: Mathieu group
zbMATH Keywords: Convay group
reviewed by: Theo Grundhöfer
MaRDI profile type: Publication
full work available at URL: jabr.1996.0132
OpenAlex ID: W2127791847
Wikidata QID: Q56872290
DOI: 10.1006/JABR.1996.0132
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description: scientific article; zbMATH DE number 883772