[GAP Forum] trival fp group

[tkohl at math.bu.edu]

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
> 
> 
>