[GAP Forum] Projection for Semidirect Product

[Alexander Hulpke]

Dear Forum,

On Oct 10, 2013, at 10/10/13 11:19, Sopsku <rrburns at cox.net> 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