A simple construction for the Fischer-Griess monster group
type: Item
instance of: scholarly article
title: A simple construction for the Fischer-Griess monster group
zbMATH Open document ID: 0564.20010
DOI: 10.1007/BF01388521
published in: Inventiones Mathematicae
publication date: 1985-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/143210
review text: In [Invent. Math. 69, 1--102 (1982; Zbl 0498.20013)] \textit{R. Griess} gave the construction of the largest, sporadic finite simple group \(F\), the Fischer-Griess monster, as the group of automorphisms of a certain 196884-dimensional algebra \(B\) in a Euclidean space. It came as a surprise that such a large group was constructed essentially without a computer. Still the construction of Griess contained extremely difficult technical details. In [ibid. 78, 491--499 (1984; Zbl 0548.20011)] and some other papers \textit{J. Tits} gave a variation of the construction of Griess. This made many computations obsolete and showed in particular in a simple way, that \(\Aut(B)\) is a finite group. In the paper under review the author gives a further variation of the construction of \(B\) and \(F\). His aim is to make computations in the algebra \(B\) as easy as possible. Griess constructs \(F\) by the centralizer \(C\) of an involution and the normalizer \(N\) of a four group. He starts with a representation of \(C\), defines a \(C\)-invariant algebra \(B\), and finds an involution \(\sigma\in N-C\), and works out in a very delicate analysis the precise shape of \(\sigma\). The author of the paper under review goes the opposite way: In a natural way he finds a fourfold cover of \(N\) as the automorphism group of a certain Moufang loop. He then works from \(N\) to \(C\). The convenient presentation of \(N\) pays off in a simplification of computations and formulas which occur when \(C\) and \(N\) are ``put together''. With the facilitation of the computations in the algebra \(B\) one also can show, that there is a certain finite number of vectors, which is invariant under \(F=\Aut(B)\), proving the finiteness of \(F\). Finally the author gives various results on subalgebras of \(B\) including some work of S. Norton.
Mathematics Subject Classification ID: 20D08
Mathematics Subject Classification ID: 20E32
zbMATH DE Number: 3899148
zbMATH Keywords: sporadic finite simple group
zbMATH Keywords: Fischer-Griess-monster
zbMATH Keywords: group of automorphisms
zbMATH Keywords: 196884-dimensional algebra
zbMATH Keywords: cover
zbMATH Keywords: Moufang loop
reviewed by: Ulrich Dempwolff
MaRDI profile type: Publication
Wikidata QID: Q55880355
cites work: The friendly giant
cites work: Q5652865
cites work: A natural representation of the Fischer-Griess Monster with the modular function J as character
OpenAlex ID: W2084539127
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description: scientific article; zbMATH DE number 3899148