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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
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Abstract

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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

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