On Coxeter's families of group presentations

Abstract In 1939 Coxeter published three infinite families of group presentations. He studied their properties, in particular determining when groups defined by members of the families are infinite and the structure of finite ones. Eight presentations remained for which the finiteness question was unsettled. We show that two of these eight presentations define finite groups (for which we give comprehensive proofs and provide detailed structural information) and that two of the presentations define infinite groups. Our results rely on substantial amounts of computer calculations, in particular on coset enumeration to prove finiteness and on computation of automatic structures using Knuth–Bendix rewriting to prove infiniteness.

[2]  Martin Edjvet,et al.  The groups Gm,n,p , 2008 .

[3]  H. S. M. Coxeter,et al.  The abstract groups , 1939 .

[4]  John J. Cannon,et al.  The Magma Algebra System I: The User Language , 1997, J. Symb. Comput..

[5]  D. Robinson.  A Course in the Theory of Groups , 1982 .

[6]  E. O'Brien,et al.  Handbook of Computational Group Theory , 2005 .

[7]  D. Holt,et al.  Computing with Abelian Sections of Finitely Presented Groups , 1999 .

[8]  George Havas,et al.  On one-relator quotients of the modular group , 2011 .

[9]  Derek F. Holt.  The Warwick automatic groups software , 1994, Geometric and Computational Perspectives on Infinite Groups.

[10]  Charles C. Sims,et al.  Computation with finitely presented groups , 1994, Encyclopedia of mathematics and its applications.

[11]  George Havas,et al.  Experiments in coset enumeration , 2001 .