Behind and beyond a theorem on groups related to trivalent graphs

George Havas, Edmund F. Robertson, Dale C. Sutherland

Research output: Contribution to journalArticlepeer-review


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.

Original languageEnglish
Pages (from-to)323-332
Number of pages10
JournalJournal of the Australian Mathematical Society
Issue number3
Publication statusPublished - Dec 2008


  • Finitely presented groups
  • Proofs
  • Todd-Coxeter coset enumeration
  • Trivalent graphs


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