Presentations of the Fischer groups \(D_ 4(2):{\mathbf S}_ 3\) and \(D_ 4(3):{\mathbf S}_ 3\)
type: Item
instance of: scholarly article
title: Presentations of the Fischer groups \(D_ 4(2):{\mathbf S}_ 3\) and \(D_ 4(3):{\mathbf S}_ 3\)
zbMATH Open document ID: 0834.20032
author: François Zara
published in: Journal of Algebra
publication date: 1996-03-20T00:00:00Z
review text: The author studies so called reduced Fischer pairs \((G,D)\), where \(G\) is a finite group generated by a conjugacy class \(D\) of 3-transpositions such that the largest solvable normal subgroup of \(G\) is in \(Z(G)\). Let \(S=\langle S\cap D\rangle\cong S_5\) be a subgroup of \(G\). Then he shows that \((G,D)\) is symmetric or classical if and only if for each \(d\in D\) there is some \(S\) such that \(D_d\cap S\neq\phi\). This suggests that the sporadic groups and the groups of the title are factors of the group \(G_0(p,q)\) generated by \(S\) and an element \(y\) with the relations \(y^2=(dy)^3=1\) for all transpositions \(d\) in \(S_5\) and \((z(25))^p=(z(15))^q\) with \(z=y(4321) y(1234)y\). The main result is that \(G_0(2,3)\cong D_4(2):S_3\), \(G_0(3,3)\cong D_4(3):S_3\). There is a similar result due to \textit{J. I. Hall} and \textit{L. Soicher} [Commun. Algebra 23, 2517-2559 (1995; Zbl 0830.20052)]. But in contrast to that paper the proof given by the author is free of any use of a computer.
Mathematics Subject Classification ID: 20F05
Mathematics Subject Classification ID: 20D05
zbMATH DE Number: 705002
zbMATH Keywords: conjugacy class of 3-transpositions
zbMATH Keywords: reduced Fischer pairs
zbMATH Keywords: finite groups
zbMATH Keywords: solvable normal subgroups
zbMATH Keywords: sporadic groups
zbMATH Keywords: relations
reviewed by: Gernot Stroth
MaRDI profile type: Publication
full work available at URL: jabr.1994.1362
OpenAlex ID: W2053057422
DOI: 10.1006/JABR.1994.1362
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description: scientific article; zbMATH DE number 705002