[GAP Forum] Projection for Semidirect Product

[Sopsku]

Alexander Hulpke <hulpke at ...> writes:

> 
> 
> Dear Forum,
> 
> On Oct 10, 2013, at 10/10/13 11:19, Sopsku <rrburns at ...> wrote:
> 
> > Dear forum,
> > 
> > I am having some difficulty understanding projections and semidirect
products. 
> > Now I would like to do soemthing similar for a semidirect product group, e.g
> > 
> > a:=AutomorphismGroup(g);
> > s:=SemidirectProduct(a,g);
> > 
> > but now
> > Projection(s,1); 
> > fails.
> > 
> > Can I use the GAP Projections to do a decomposition similar to the  direct
> > product example above?
> 
> According to the manual, for a semidirect product N:S
> Projection(s) 
> returns the projection onto S, there is no projection onto N which is a
group homomorphism (and thus no
> numeric parameter to Projection).
> 
> If you want to get an N-part of a product element g, you could divide off
the canonical representative for the
> projection image, for example:
> 
> PreImagesRepresentative(Embedding(s,1),
g/Image(Embedding(s,2),Image(Projection(s),g));
> 
> Regards,
> 
>     Alexander Hulpke
> 

Sorry for being so dense, but I do not fully understand the command and it
fails so I am not quite sure what Prof. Hulple intended. I did try the
following:

npart1:=function(elm)
   return Image(e1,Image(p,elm))^p;
end;

npart2:=function(elm)
   return PreImagesRepresentative(e2,elm/Image(e1,Image(p,elm)));
end;

PrintArray(List(Elements(S),e->[npart1(e),npart2(e)]));

Which is more or less what I think I am looking for. Is this kind of what
Prof. Hulpke intended.

Again thank you for any help or comments.
    Ron