Hostname: page-component-78c5997874-s2hrs Total loading time: 0 Render date: 2024-11-09T01:37:25.732Z Has data issue: false hasContentIssue false
  • English
  • Français

Some challenging group presentations

Part of: Special aspects of infinite or finite groups

Published online by Cambridge University Press:  09 April 2009

George Havas ,
Derek F. Holt ,
P. E. Kenne  and
Sarah Rees
George Havas
Affiliation:
Centre for Discrete Mathematics and Computing Computer Science and Electrical Engineering University of Queensland, Qld 4072Australia e-mail: [email protected]
Derek F. Holt
Affiliation:
Mathmatics Institute University of Warwick Conventry CV4 7AL UK e-mail: [email protected]
P. E. Kenne
Affiliation:
Electrical and Electronic Engineering The University of Adelaide SA 5005 Australia e-mail: [email protected]
Sarah Rees
Affiliation:
Mathematics and Statistics University of Newcastle Newcastle upon Tyne NEI 7RU UK e-mail: [email protected]
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

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.

MSC classification

Secondary: 20F05: Generators, relations, and presentations
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 67 , Issue 2 , October 1999 , pp. 206 - 213
Copyright
Copyright © Australian Mathematical Society 1999

References

[1]Bosma, W., Cannon, J. and Playoust, C., ‘The MAGMA algebra system I: The user language’, J. Symbolic Comput. 24 (1997), 235–265.CrossRefGoogle Scholar
[2]Brückner, H., Algorithmen für endliche auflösbare Gruppen und Anwendungen (Ph.D. Thesis, Aachener Beiträge zur Mathematik, Band 22, Verlag der Augudtinus Buchhandlung, 1998).Google Scholar
[3]Campbell, C. M. and Robertson, E. F., ‘Presentations for the simple groups G, 105 < |G| < 106’, Comm. Algebra 12 (1984), 2643–2663.CrossRefGoogle Scholar
[4]Cannon, J. and Havas, G., ‘Algorithms for groups’, Austral. Compouter Journal 24 (1992), 51–60.Google Scholar
[5]Epstein, D. B. A., Holt, D. F. and Rees, S. E., ‘The use of Knuth-Bendix methods to solve the word problem in automatic groups’, J. Symbolic Comput. 12 (1991), 397–414.CrossRefGoogle Scholar
[6]Gilman, R. H., ‘Presentations of groups and monoids’, J. Algebra 57 (1979), 544–554.CrossRefGoogle Scholar
[7]Group-pub-forum e-mail list, , http://www2.bath.ac.uk/~masgcs/problem/problem11.html, Problem 11, 1996.Google Scholar
[8]Havas, G., ‘Coset enumeration strategies’, in: Proceedings of the 1991 International Symposium on Symbolic and Algebraic Computation (ACM Press, New York, 1991) pp. 191–199.CrossRefGoogle Scholar
[9]Havas, G., Newman, M. F. and O'Brien, E. A., ‘Groups of deficiency zero’, in: Geometric and computational perpectives on infinite groups, DIMACS Series in Discrete Mathematics and Theoretical Computer Science 25 (1996) pp. 53–67.Google Scholar
[10]Havas, G. and Ramsay, C., Coset enumeration: ACE version 3 (1999), Available as http://www.csee.uq.edu.au/~havas/ace3.tar.gz.Google Scholar
[11]Havas, G. and Ramsay, C., ‘Experiments in coset enumeration’, Centre for Discrete Mathmatics and Computing Report 13 (The University of Queensland, 1999).Google Scholar
[12]Holt, D. F., ‘An alternative proof that the Fibonacci group F(2, 9) is infinite’, Experiment. Math. 4 (1995), 97–100.CrossRefGoogle Scholar
[13]Holt, D. F., KBMAG Knuth-Bendix in monoids and automatic groups, software package (1996), Available from ftp.warwick.maths.ac.ukindirectorypeople/dfh/kbmag2.Google Scholar
[14]Holt, D. F. and Hurt, D. F., ‘Automatic coset systems and subgroup presentations’, J. Symbolic Comput. 27 (1999), 1–19.CrossRefGoogle Scholar
[15]Johnson, D. L. and Robertson, E. F., ‘Finite groups of deficiency zero’, in: Homological group theory, London Math. Soc. Lecture Note Ser. 36 (Cambridge University Press, Cambridge, 1979) pp. 275–289.CrossRefGoogle Scholar
[16]Kenne, P. E., ‘Some new efficient soluble groups’, Comm. Algebra 18 (1990), 2747–2753.CrossRefGoogle Scholar
[17]Kenne, P. E., Minimal group presentations: a computational approach (Ph.D. Thesis, Australian National University, 1991).Google Scholar
[18]Knuth, D. E. and Bendix, P. B., ‘Simple word problems in elementary algebra’, in: Computational problems in abstract algebra (Pergamon Press, Oxford, 1970) pp. 263–297.Google Scholar
[19]Leech, J., ‘Coset enumeration on digital computers’, Proc. Camb. Phil. Soc. 59 (1963), 257–267.CrossRefGoogle Scholar
[20]Mazurov, V. D. and Khukhro, E. I. (eds.), The Kourovka notebook: unsolved problems on group theory, 13th Edition (Akad. Nauk SSSR Sibirsk.Otdel., Inst. Mat., Novosibirsk, 1995).Google Scholar
[21]Mendelsohn, N. S., ‘An algorithmic solution for a word problem in group theory’, Canad. J. Math. 16 (1964), 509–516 Correction, Canad. J. Math. 17 (1965) 505.CrossRefGoogle Scholar
[22]Moore, E. H., ‘Concerning the abstract groups of order k! abd ½k! holohedrically isomorphic with the symmetric and the alternating substitution-groups on k letters’, Proc. London Math. Soc. 28 (1897), 357–366.Google Scholar
[23]Plesken, W., ’, J. Symbolic Comput. 4 (1987), 111–122.CrossRefGoogle Scholar
[24]Schönert, M. et al. , GAP – Groups, algorithms, and programming, Lehrstuhl D für Mathematik, 5th Edition (Rheinisch Westfälische Technische Hochschule, Aachen, Germany, 1995).Google Scholar
[25]Sims, C. C., Computation with finitely presented groups, Encyclopedia of Mathematics 48 (Cambridge University Press, Cambridge, 1994).Google Scholar
[26]Todd, J. A. and Coxeter, H. S. M., ‘A practical method for enumerating cosets of finite abstract groups’, Proc. Edinburgh Math. Soc. 5 (1936), 26–34.CrossRefGoogle Scholar