[GAP Forum] Simplifying presentations of abelian groups

[Alexander Hulpke]

Dear GAP Forum,

Martin Jones wrote:

>  I have an
> finitely presented abelian group and I want to obtain a simple
> presentation for the group. The command SimplifedFpGroup is not  
> working
> satisfactorily (possibly because it doesn't introduce new generators).
> How can I make GAP use the fact that the group is abelian and use an
> appropriate method?

I would write the presentation in matrix form and then use the  
transforming matrices for the smith normal form:
The command is `SmithNormalFormIntegerMatTransforms'.

Incidentally this is done by the existing command  
`MaximalAbelianQuotient' and (if the group is infinite -- otherwise  
the range is a pc group and the presentation a bit mixed up) you might  
find the range group of this command more to your liking. In your  
example, e.g. I get:

gap> ma:=MaximalAbelianQuotient(Gsimp);
[ f1, f2, f5, f6 ] -> [ f2, f1*f2^-1, f3, f4 ]
gap> RelatorsOfFpGroup(Range(ma));
[ f1^-1*f2^-1*f1*f2, f1^-1*f3^-1*f1*f3, f1^-1*f4^-1*f1*f4,  
f2^-1*f3^-1*f2*f3,
   f2^-1*f4^-1*f2*f4, f3^-1*f4^-1*f3*f4, f1^4 ]

(You could duplicate and modify the code to always return the result  
as a finitely presented group).

Best,

    Alexander Hulpke