Constructing matrix representations of finitely presented groups
type: Item
instance of: scholarly article
title: Constructing matrix representations of finitely presented groups
zbMATH Open document ID: 0789.20002
DOI: 10.1016/S0747-7171(08)80095-8
author: S. Singh
published in: Journal of Symbolic Computation
publication date: 1992-09-27T00:00:00Z
review text: Most decision questions that can be asked about a finitely presented (f.p.) group, e.g. the question if it is finite or infinite, are algorithmically unsolvable. Therefore computational methods are of great interest that either compute certain factor groups of an f.p. group (e.g. its commutator factor group or its largest nilpotent group of preassigned class) or that can verify properties, e.g. finiteness, i.e. that eventually give the order of an f.p. group provided this group is finite. The Todd-Coxeter method for the enumeration of the cosets of a subgroup \(U\) of an f.p. group \(G\) (for which \(U\) is given by a finite set of generators of \(U\), qua words in the generators of \(G\)) can be used either way; if \(U\) is chosen to be the trivial subgroup it is a verification method for \(| G|\), if \(U\) is chosen nontrivial of finite index in \(G\) it provides the permutation representation of \(G\) on the cosets of \(U\) and hence a finite factor group of \(G\). In analogy the present paper provides a method for the determination of a matrix representation over a field \(F\) of an f.p. group \(G\). Such a matrix representation yields of course a representation of the group algebra \(FG\), and in fact the method described can be more generally applied to finitely presented algebras. The first chapter clarifies underlying ideas and the relation of this new method, the idea of which has been suggested by J. Conway and L. Soicher, to the Todd-Coxeter method. The second chapter gives details of the algorithm. Fortunately the author does not only give a good description of the method but has implemented it with consideration for practical efficiency. This implementation is described in Chapter 3, it is meanwhile e.g. available in the form of a share library of the GAP system. In chapter 4 some results obtained with this algorithm are given. The reviewer can add that meanwhile Linton's program has become a key part in the implementation of a new Soluble Quotient Algo…
Mathematics Subject Classification ID: 20-04
Mathematics Subject Classification ID: 20F05
Mathematics Subject Classification ID: 20C40
Mathematics Subject Classification ID: 20F10
Mathematics Subject Classification ID: 68W30
zbMATH DE Number: 60657
zbMATH Keywords: finitely presented groups
zbMATH Keywords: enumeration of cosets
zbMATH Keywords: soluble quotient algorithm
zbMATH Keywords: computational group theory
zbMATH Keywords: computational methods
zbMATH Keywords: factor groups
zbMATH Keywords: finiteness
zbMATH Keywords: Todd-Coxeter method
zbMATH Keywords: finite set of generators
zbMATH Keywords: permutation representation
zbMATH Keywords: matrix representation
zbMATH Keywords: algorithm
zbMATH Keywords: efficiency
zbMATH Keywords: implementation
reviewed by: Joachim Neubüser
describes a project that uses: GAP
MaRDI profile type: Publication
cites work: Implementation and Analysis of the Todd-Coxeter Algorithm
cites work: Q3684278
cites work: Q5727698
cites work: Q3223036
cites work: Double coset enumeration
cites work: Q3950713
cites work: Q4145843
cites work: A practical method for enumerating cosets of a finite abstract group
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description: scientific article; zbMATH DE number 60657