Hostname: page-component-848d4c4894-wg55d Total loading time: 0 Render date: 2024-04-30T18:31:15.890Z Has data issue: false hasContentIssue false
  • English
  • Français

BEHIND AND BEYOND A THEOREM ON GROUPS RELATED TO TRIVALENT GRAPHS

Part of: Group theory and generalizations Special aspects of infinite or finite groups

Published online by Cambridge University Press:  01 December 2008

GEORGE HAVAS ,
EDMUND F. ROBERTSON  and
DALE C. SUTHERLAND
GEORGE HAVAS*
Affiliation:
ARC Centre for Complex Systems, School of Information Technology and Electrical Engineering, The University of Queensland, Queensland 4072, Australia (email: havas@itee.uq.edu.au)
EDMUND F. ROBERTSON
Affiliation:
School of Mathematics and Statistics, University of St Andrews, North Haugh, St Andrews, Fife KY16 9SS, Scotland (email: efr@st-andrews.ac.uk)
DALE C. SUTHERLAND
Affiliation:
School of Mathematics and Statistics, University of St Andrews, North Haugh, St Andrews, Fife KY16 9SS, Scotland
*
✉For correspondence; e-mail: havas@itee.uq.edu.au
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.

In 2006 we completed the proof of a five-part conjecture that was made in 1977 about a family of groups related to trivalent graphs. This family covers all 2-generator, 2-relator groups where one relator specifies that a generator is an involution and the other relator has three syllables. Our proof relies upon detailed but general computations in the groups under question. The proof is theoretical, but based upon explicit proofs produced by machine for individual cases. Here we explain how we derived the general proofs from specific cases. The conjecture essentially addressed only the finite groups in the family. Here we extend the results to infinite groups, effectively determining when members of this family of finitely presented groups are simply isomorphic to a specific quotient.

Keywords

finitely presented groupsproofsTodd–Coxeter coset enumerationtrivalent graphs

MSC classification

Secondary: 20F05: Generators, relations, and presentations 20-04: Explicit machine computation and programs (not the theory of computation or programming)
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 85 , Issue 3 , December 2008 , pp. 323 - 332
Copyright
Copyright © Australian Mathematical Society 2009

Footnotes

This research was partially supported by the Australian Research Council and by the Engineering and Physical Sciences Research Council grant EP/C523229/01, ‘Multidisciplinary Critical Mass in Computational Algebra and Applications’.

References

[1]Bosma, W., Cannon, J. and Playoust, C., ‘The Magma algebra system I: the user language’, J. Symbolic Comput. 24 (1997), 235–265. See also http://www.maths.usyd.edu.au:8000/u/magma/.CrossRefGoogle Scholar
[2]Campbell, C. M., Coxeter, H. S. M. and Robertson, E. F., ‘Some families of finite groups having two generators and two relations’, Proc. Roy. Soc. Lond. Ser. A 357 (1977), 423–438.Google Scholar
[3]Campbell, C. M. and Robertson, E. F., ‘On 2-generator 2-relation soluble groups’, Proc. Edinb. Math. Soc. (2) 23 (1980), 269–273.CrossRefGoogle Scholar
[4]Campbell, C. M. and Robertson, E. F., ‘Groups related to F a,b,c involving Fibonacci numbers’, in: The Geometric Vein (Springer, New York, 1981), pp. 569–576.CrossRefGoogle Scholar
[5] The GAP Group, GAP   Groups, Algorithms, and Programming, Version 4.4, 2004. See also http://www.gap-system.org/.Google Scholar
[6]Havas, G. and Ramsay, C., ‘On proofs in finitely presented groups’, in: Groups, St Andrews, 2005, London Mathematical Society Lecture Note Series, 340 (Cambridge University Press, Cambridge, 2007), pp. 457–474.CrossRefGoogle Scholar
[7]Havas, G. and Robertson, E. F., ‘The F a,b,c conjecture, I’, Irish Math. Soc. Bull. 56 (2005), 75–80.CrossRefGoogle Scholar
[8]Havas, G., Robertson, E. F. and Sutherland, D. C., ‘The F a,b,c conjecture is true, II’, J. Algebra 300 (2006), 57–72.CrossRefGoogle Scholar
[9]Leech, J., ‘Computer proof of relations in groups’, in: Topics in Group Theory and Computation (Academic Press, New York, 1977), pp. 38–61.Google Scholar
[10]Ramsay, C., PEACE 1.100: Proof Extraction After Coset Enumeration, Technical Report 22, Centre for Discrete Mathematics and Computing, The University of Queensland, 2003.Google Scholar
[11]Sutherland, D. C., ‘Computer-assisted proofs and the F a,b,c conjecture’, PhD Thesis, University of St Andrews, 2005.Google Scholar