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Some challenging group presentations

Published online by Cambridge University Press:  09 April 2009

George Havas
Affiliation:
Centre for Discrete Mathematics and Computing Computer Science and Electrical Engineering University of Queensland, Qld 4072Australia e-mail: havas@csee.uq.edu.au
Derek F. Holt
Affiliation:
Mathmatics Institute University of Warwick Conventry CV4 7AL UK e-mail: dfh@maths.warwick.ac.uk
P. E. Kenne
Affiliation:
Electrical and Electronic Engineering The University of Adelaide SA 5005 Australia e-mail: pek@dvcr.adelaide.edu.au
Sarah Rees
Affiliation:
Mathematics and Statistics University of Newcastle Newcastle upon Tyne NEI 7RU UK e-mail: Sarah.rees@newcastle.ac.uk
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Abstract

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We study some challenging presentations which arise as groups of deficiency zero. In four cases we settle finiteness: we show that two presentations are for finite groups while two are for infinite groups. Thus we answer three explicit questions in the literature and we provide the first published deficiency zero presentation for a group with derived length seven. The tools we use are coset enumeration and Knuth-Bebdix rewriting, which are well-established as methods for proving finiteness or otherwise of a finitely presented group. We briefly comment on their capabilities and compare their performance.

Type
Research Article
Copyright
Copyright © Australian Mathematical Society 1999

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