[GAP Forum] ReReidemeister Schreier method and Homomorphism to the symmetric group

[Alexander Hulpke]

Dear GAP Forum,

On Mar 15, 2007, at 1:50 AM, Michael Fridman wrote:
> I have a question regarding the Reidemeister-Schreier method. I have a
> finitely presented group G and a set of the schreier generators of a
> subgroup H. so using "PresentationSubgroup " I found a presentation  
> of the
> subgroup H. However, I need to know how the generators of the  
> subgroup H
> (when they are calculated using the RMS method) are expressed in  
> terms of
> the generators of G. how can do it?

The easiest way to do so is to use `IsomorphismFpGroup' applied to H.  
This will do a Reidemeister-Schreier rewriting (and Tietze  
transformations) to obtain a presentation for H. The `Range' of this  
homomorphism is the newly presented group. You can use the  
homomorphism to rewrite elements between H<=G and the new group  
isomorphic to H.
(If you wanted to do this with PresentationSubgroup you would have to  
do the translation by hand. See the code in lib/ghomfp.gi.)

He also asked:

> Given a finitely presented group G, I need
> to find all the surjective homomorphisms to the symmetric group of n
> elements (when n  is given). Can I use GAP to do it?

I assume you want one n at a time. You could use `GQuotients' to do  
this. (However be aware that this is likely to fail if your n gets  
too big. n=10 should be doable. n=100 is likely out of range.)

Best,

     Alexander Hulpke


-- Colorado State University, Department of Mathematics,
Weber Building, 1874 Campus Delivery, Fort Collins, CO 80523-1874, USA
email: hulpke at math.colostate.edu, Phone: ++1-970-4914288
http://www.math.colostate.edu/~hulpke