Dear George,
Thank you very much. One of examples is
F:=FreeGroup(3);x:=F.1;y:=F.2;z:=F.3;a:=x*y;b:=x*z;c:=y*z;
R:=[x^2,y^2,z^2,a^4,b^4,c^4,
(a^2*z)^4,(b^2*y)^4,(c^2*x)^4,
(a^2*b^2)^4,(a^2*c^2)^4,(b^2*c^2)^4,(x*y*z*y)^4,(x*z*y*z)^4,(z*x*y*x)^4];
G:=F/R;
w:=(x*y*z*x*z*y*x*y*z*x*z)^2;
Best wishes, Victor.
2011/6/1 GH UQ <ha...@itee.uq.edu.au>
> 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
>
--
Victor Danilovich Mazurov
Institute of Mathematics
Novosibirsk 630090
Russia
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