Computer-assisted proofs and the F[super](a,b,c) conjecture

Abstract

This thesis studies finitely presented groups and the process known as coset enumeration, which finds the index of a finitely generated subgroup in a finitely presented group, provided this index is finite. The Todd-Coxeter algorithm for coset enumeration is described, as well as its modified version, additionally finding a presentation for the subgroup. Coset enumeration is suitable for computer implementation, and GAP and ACE, two programs containing such functions using different strategies, are outlined. Proof Extraction After Coset Enumeration (PEACE) is a computer pro¬ gram that allows one to show a group element is in the subgroup. Descriptions are provided of modifications to PEACE, giving this program the extra functionality of creating subgroup presentations with the Modified ToddCoxeter algorithm. Using different strategies during the enumeration to determine varied subgroup presentations is also discussed. Additionally, a program converting the output of the original PEACE program, showing an element's membership of the subgroup, into a lemma-based step by step proof is implemented and described. 'The Fᵃᵇᶜ conjecture' was proposed by Campbell, Coxeter and Robertson in 1977 to classify the groups Fᵃ,ᵇ,ᶜ =〈r,s|r²,rsᵃrsᵇrsᶜ〉 By considering the homomorphic image Hᵃᵇᶜ=〈r,s|r²,rsᵃrsᵇrsᶜ,s²⁽ᵃᵇᶜ⁾〉The lemma-based proof generating program is used as an aid in considering the groups Fᵃ,ᵇ,ᶜ and the corresponding conjecture. Lastly, a proof showing this conjecture to be true is provided.