[GAP Forum] Fp group

[GH UQ]

On Wed, 1 Jun 2011, Victor D. Mazurov wrote:

> Suppose that GAP shows (using coset enumeration algorithm) that a group
> $G=<F| R>$ where $F$ is a finitely generated free group and $R$ is a finite
> set of words in $F$ is finite. Let $w\in F$ such that the image of  $w$ in
> $G$ is 1.  How one can find (using GAP) some $r_1,...,r_m\in (R\cup R^{-1})$
> and $t_1,...,t_m\in F$ such that $w=r_1^{t_1}\cdots r_m^{t_m}$? Best wishes,
> V.D. Mazurov
> 
In 2006 Dale Sutherland developed a Gap package for PEACE (proof
extraction after coset enumeration) which can do this.  PEACE is described
in "On proofs in finitely presented groups" by George Havas and Colin
Ramsay, Groups St Andrews 2005, Volume II, London Mathematical Society
Lecture Note Series 340, Cambridge University Press (2007), 475-485.

Applications of PEACE appear in various places including:
Dale's PhD Thesis (St Andrews, 2006);
"The Fa,b,c conjecture is true, II" by George Havas, Edmund F. Robertson
and Dale C. Sutherland, Journal of Algebra 300 (2006), 57-72;
"Andrews-Curtis and Todd-Coxeter proof words" by George Havas and Colin
Ramsay, Groups St Andrews 2001 in Oxford, Volume I, London Mathematical
Society Lecture Note Series 304, Cambridge University Press (2003)
232-237.

The GAP package has not been generally released (as far as I know).
I have a stand-alone version which I am happy to consider trying to
use to solve specific problems, if desired.

Best wishes...  George Havas   http://www.itee.uq.edu.au/~havas