[GAP Forum] trival fp group

[dmitrii.pasechnik at cs.ox.ac.uk]

Dear Tim,

you might have better luck with GAP's package ACE, perhaps it print out
a sufficiently detailed trace of what it does during building and reducing the coset table.
https://www.gap-system.org/Manuals/pkg/ace-5.2/htm/CHAP001.htm
Also, on the webpage of one of the authors of ACE http://staff.itee.uq.edu.au/havas/
one can find

Proof Extraction After Coset Enumeration: PEACE version 1
Colin Ramsay's source code + some test scripts

Another appoach might be to try a Knuth-Bendix rewriting on your
presentation, perhaps it will be more illuminating.
https://www.gap-system.org/Packages/kbmag.html
(I don't know anything about this approach, whether it will work
for this presentation or not, I have no idea)

HTH
Dima


On Thu, Sep 19, 2019 at 03:14:59PM -0400, tkohl at math.bu.edu wrote:
> 
> Dear Alexander, 
> 
> Thank you very much for the explanation.
>  
> 	-Tim
> 
> On Thu, 19 Sep 2019, Hulpke,Alexander wrote:
> 
> > Dear GAP Forum,
> > 
> > > On Sep 19, 2019, at 10:56 AM, tkohl at math.bu.edu wrote:
> > > I was looking at the finitely presented group
> > > 
> > > G=<x,y | yx^3=x^2y; y^3x=xy^2>
> > > 
> > > where one can show that G=[G,G], which is pretty easy.
> > > 
> > > I had a nagging suspicion that it is actually trivial
> > > and in GAP I found that this was the case:
> > > 
> > > gap> f := FreeGroup( "x", "y" );; 
> > > gap> g := f / [ f.2*f.1^3*f.2^(-1)*f.1^(-2),f.2^3*f.1*f.2^(-2)*f.1^(-1) ];
> > > gap> Size(g);
> > > 1
> > > 
> > > And I was able to work out that this was indeed the case by
> > > playing with the relations.
> > > 
> > > What I'm wondering is whether I can make GAP show me
> > > how it determined this group was trivial?
> > 
> > What GAP does is to try coset enumeration by a cyclic subgroup, and -- assuming this enumeration terminates and returns the index -- rewrite the presentation to this cyclic subgroup, calculating the order of the subgroup then is easy.
> > 
> > There is no mechanism provided to extract a proof from this. You would have to interface rather seriously with the coset enumeration (i.e. basically rewrite the routine) to extract an actual proof.
> > You might want to look at
> > 
> > http://dx.doi.org/doi:10.1017/S0004972700018529
> > 
> > for a description on how this could be done (it is not implemented in GAP).
> > 
> > Best,
> > 
> >   Alexander Hulpke
> > 
> > -- Colorado State University, Department of Mathematics,
> > Weber Building, 1874 Campus Delivery, Fort Collins, CO 80523-1874, USA
> > email: hulpke at colostate.edu
> > http://www.math.colostate.edu/~hulpke
> > 
> > 
> > 
> 
> _______________________________________________
> Forum mailing list
> Forum at gap-system.org
> https://mail.gap-system.org/mailman/listinfo/forum