[GAP Forum] trival fp group

[Hulpke,Alexander]

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