[GAP Forum] Simplifying presentations of abelian groups
hulpke at math.colostate.edu
Mon Apr 21 20:51:59 BST 2008
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
> 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:
[ f1, f2, f5, f6 ] -> [ f2, f1*f2^-1, f3, f4 ]
[ f1^-1*f2^-1*f1*f2, f1^-1*f3^-1*f1*f3, f1^-1*f4^-1*f1*f4,
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).
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