Presentations of Fischer groups. II
type: Item
instance of: scholarly article
title: Presentations of Fischer groups. II
zbMATH Open document ID: 0780.20021
DOI: 10.1007/BF01264517
author: Marguerite-Marie Virotte-Ducharme
published in: Geometriae Dedicata
publication date: 1993-05-16T00:00:00Z
review text: [Part I cf. ibid. 41, No. 3, 275-335 (1992; Zbl 0757.20004).] A conjugacy class \(D\) of involutions of a group \(G\) such that the product of any two distinct elements of \(D\) has order 2 or 3 is called a class of 3-transpositions. If \(D\) generates \(G\), then \((G,D)\) is called a Fischer pair and \(D\) is called a Fischer class of \(G\). The finite groups \(G = Fi_{22},Fi_{23},Fi_{24}\) discovered by B. Fischer are each generated by a class of 3-transpositions \(D\). This article gives, for each of these three Fischer groups \(G\), a presentation \((X,R)\) where \(X\) is contained in \(D\) and \(R\) is a set of relations on \(X\) and a subset \(R_ F\) such that \((X,R_ F)\) presents a Coxeter group. Let \(H\) denote the centralizer of an element \(d\) of \(D\) where \(D\) is the Fischer class of \(G=Fi_{24}\) (resp. \(Fi_{23},Fi_{22}\)). Then \(H\) is a central extension of \(Fi_{23}\) (resp. \(Fi_{22}\), \(PSU(6,4)\)) and \(H\) has a presentation \((X_ 0,R_ 0)\) where \(X_ 0\) is contained in the Fischer class of \(H\). The desired construction proceeds by demonstrating the existence of an element \(b\) in \(D\) such that \(X = X_ 0 \cup \{b\}\) generates \(G\) and then showing that the existing theory can be applied to a set \(R\) of relations on \(X\).
Mathematics Subject Classification ID: 20F05
Mathematics Subject Classification ID: 20D08
zbMATH DE Number: 166875
zbMATH Keywords: conjugacy class of involutions
zbMATH Keywords: class of 3-transpositions
zbMATH Keywords: Fischer pair
zbMATH Keywords: finite groups
zbMATH Keywords: Fischer groups
zbMATH Keywords: presentation
zbMATH Keywords: relations
zbMATH Keywords: centralizer
zbMATH Keywords: central extension
reviewed by: Morton E. Harris
MaRDI profile type: Publication
cites work: Q4740746
cites work: Q5830680
cites work: Q3684278
cites work: \(3\)-transposition groups of orthogonal type
cites work: Q3848237
cites work: Finite groups generated by 3-transpositions. I
cites work: Schur Multipliers of Finite Simple Groups of Lie Type
cites work: Schur multipliers of some sporadic simple groups
cites work: Une construction du groupe de Fischer $Fi(24)$
cites work: Presentations of Fischer groups. I
cites work: Présentation de certains couples fischériens de type classique
full work available at URL: https://doi.org/10.1007/bf01264517
OpenAlex ID: W3004302205
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description: scientific article; zbMATH DE number 166875