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Proving a group trivial made easy: A case study in coset enumeration

Published online by Cambridge University Press:  17 April 2009

George Havas  and
Colin Ramsay
George Havas
Affiliation:
Centre for Discrete Mathematics and Computing, Department of Computer Science and Electrical Engineering, The University of Queensland, Queensland 4072Australia, e-mail: havas@csee.uq.edu.aucram@csee.uq.edu.au
Colin Ramsay
Affiliation:
Centre for Discrete Mathematics and Computing, Department of Computer Science and Electrical Engineering, The University of Queensland, Queensland 4072Australia, e-mail: havas@csee.uq.edu.aucram@csee.uq.edu.au
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Abstract

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Coset enumeration, based on the methods described by Todd and Coxeter, is one of the basic tools for investigating finitely presented groups. The process is not well understood, and various pathological presentations of, for example, the trivial group have been suggested as challenge problems. Here we consider one such family of presentations proposed by B.H. Neumann. We show that the problems are much easier than they first appear, albeit at the expense of considerable preliminary ‘experimentation’. This demonstrates how far the range of applicability of coset enumeration has improved.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 62 , Issue 1 , August 2000 , pp. 105 - 118
Copyright
Copyright © Australian Mathematical Society 2000

References

[1]Cannon, J.J., Dimino, L.A., Havas, G. and Watson, J.M., ‘Implementation and analysis of the Todd-Coxeter algorithm’ Math. Comp. 27 (1973), 463–490.CrossRefGoogle Scholar
[2]Cooperman, G. and Havas, G., ‘Practical Parallel coset enumeration’, in Workshop on High Performance Computing and Gigabit Local Area Networks, (Cooperman, G., Michler, G. and Vinck, H., Editors), Lecture Notes in Control and Information Sciences 226 (Springer-Verlag, Berlin, Heidelberg, New York, 1997), pp. 15–27.Google Scholar
[3]Havas, G., ‘Coset enumeration strategies’, in ISSAC'91 (Proceedings of the 1991 International Symposium on Symbolic and Algebraic Computation), (Watt, Stephen M., Editor) (ACM Press, New York, 1991), pp.191–199.Google Scholar
[4]Havas, G. and Ramsay, C., ‘ACE, Version 3 (1999)’, Available as http://www.csee.uq.edu.au/-haves/ace3.tar.gz.Google Scholar
[5]Havas, G. and Ramsay, C., ‘Experiments in coset enumeration’, (Technical Report 13, Centre for Discrete Mathematics and Computing, The University of Queensland, 1999).Google Scholar
[6]Higman, G., ‘A finitely generated infinite simple group’, J. London. Math. Soc. 26 (1951), 61–64.Google Scholar
[7]Holt, D.F., ‘KBMAG, Version 2.2 (1997)’, Available via anonymous ftp from ftp.maths.warwick.ac.uk, in directory /people/dfh/kbmag2.Google Scholar
[8]Knuth, D.E. and Bendix, P.B., ‘Simple word problems in universal algebras’, in Computational Problems in Abstract Algebra (Pergamon Press, Oxford, 1970), pp. 263–297.Google Scholar
[9]Leech, J., ‘Coset enumeration on digital computers’, Proc. Cambridge Philos. Soc. 59 (1963), 257–267.CrossRefGoogle Scholar
[10]Leech, J., ‘Coset enumeration’, in Computational Group Theory, (Atkinson, Michael D., Editor) (Academic Press, London, New York, 1984), pp. 3–18.Google Scholar
[11]Neubüser, J., ‘An elementary introduction to coset-table methods in computational group theory’, in Groups – St. Andrews 1981, London Mathematical Society Lecture Note Series 71 (Cambridge University Press, Cambridge, 1982), pp. 1–45.Google Scholar
[12]Neumann, B.H., ‘An essay on free products of groups with amalgamations’, Philos. Trans. Royal Soc. London Ser. A 264 (1954), 503–554.Google Scholar
[13]Neumann, B.H., ‘Proofs’, Math. Intelligencer 2 (1979), 18–19.Google Scholar
[14]Sims, C.C., Computation with finitely presented groups (Cambridge University Press, Cambridge, 1994).CrossRefGoogle Scholar
[15]Sims, C.C., ‘RKBP, Version 1.3 (1995)’, Available via anonymous ftp from dimacs.rutgers.edu, in directory /pub/rkbp.Google Scholar
[16]Todd, J.A. and Coxeter, H.S.M., ‘A practical method for enumerating cosets of finite abstracts groups’, Proc. Edinburgh Math. Soc. 5 (1936), 26–34.CrossRefGoogle Scholar