A construction which can be used to produce finitely presented infinite simple groups
type: Item
instance of: scholarly article
title: A construction which can be used to produce finitely presented infinite simple groups
zbMATH Open document ID: 0544.20027
DOI: 10.1016/0021-8693(84)90172-8
author: E. Scott
published in: Journal of Algebra
publication date: 1984-01-01T00:00:00Z
review text: This is the first in a series of three papers discussing finitely presented simple groups. The work is based on earlier work by \textit{R. J. Thompson} [Word problems II, Stud. Logic Found. Math. 95, 401-441 (1980; Zbl 0431.20030)] and \textit{G. Higman} [''Finitely presented simple groups'', Notes on Pure Mathematics 8, The Australian National University Canberra (1974)]. The aim is to describe a procedure which will yield a large class of finitely presented simple groups. Let W be a free monoid of rank n greater than 1. A right ideal Y of W is called inescapable if, given any u in W, there is some w in W with uw in Y. The ideal Y is called cofinite if Y is cofinite in W as subset. A function \(\mu\) on a subspace of W is called a homomorphism if \((uw)\mu =(u\mu)w\) whenever these are defined, \(\mu\) is an isomorphism if it is also bijective between its domain and co-domain. Further, \(\mu\) is inescapable (cofinite) it its domain and range are inescapable (cofinite). It is not difficult to show that every inescapable isomorphism has a unique maximal inescapable extension. The inescapable isomorphisms then form a group \({\mathcal G}_{n,1}\) with the operation being ''composition followed by formation of the maximal extension''. Similarly the cofinite isomorphisms form a subgroup \(G_{n,1}\). (The latter is amongst the groups constructed by Higman in [loc. cit].) The author constructs her groups by taking suitable subgroups H (obtained as subgroups of inverse limits of wreath products) of \({\mathcal G}_{n,1}\) and showing that the groups \(<G_{n,1},H>\) are finitely presented. She then uses a result, which she ascribes largely to Thompson that if K is a subgroup of \({\mathcal G}_{n,1}\) containing \(G_{n,1}\) then the derived group of K is simple. Since the finitely presented groups mentioned above have derived group of finite index, this yields the required class of finitely presented simple groups.
Mathematics Subject Classification ID: 20E32
Mathematics Subject Classification ID: 20F05
Mathematics Subject Classification ID: 20F10
Mathematics Subject Classification ID: 20E07
zbMATH DE Number: 3865565
zbMATH Keywords: finitely presented simple groups
zbMATH Keywords: inescapable isomorphisms
zbMATH Keywords: cofinite isomorphisms
zbMATH Keywords: inverse limits of wreath products
zbMATH Keywords: derived group of finite index
reviewed by: J. R. J. Groves
MaRDI profile type: Publication
full work available at URL: 0021-8693(84)90172-8
OpenAlex ID: W1977665851
cites work: Subgroups of finitely presented groups
cites work: The embedding of certain linear and Abelian groups in finitely presented simple groups
cites work: Q3867972
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description: scientific article; zbMATH DE number 3865565