[GAP Forum] elements of wreath products?

[Steve Linton]

It seems to me we need a slightly more general function than Projection. 

Maybe something like Components(W ,w) returning a list of n+1 elements of
suitable subgroups. 

Alternatively, and perhaps cleaner, one could provide a way to get at the
subgroup of W which is a direct product with its Projections and Embeddings.

	Steve

On Tue, 11 Sep 2007 14:50:04 +0200
Burkhard Höfling <burkhard at hoefling.name> wrote:

> Dear Keith, dear all,
> 
> >> What exists for a wreath product as described is
> >> Projection(W);  # no index!
> >> which is the projection onto P,
> >> Embedding(W,i) # i=1..n
> >> the homomorphism G->W giving the i-th copy of G and
> >> Embedding(W,n+1)
> >> giving the complement P to G^n.
> >>
> >> To get the i-th component of an element x thus one needs to split  
> >> off-
> >> the p-part first and then use the pre-image under a suitable  
> >> embedding:
> >>
> >> PreImagesRepresentative(Embedding(W,i),x/Image(Embedding(W,n+1),Image
> >> (Projection(W),x)));
> >
> > Laurent Bartholdi had already suggested that I try
> >
> >   PreImagesRepresentative(Embedding(W,i),w)
> >
> > which seems to work.  However, the manual seems to suggest that this
> > shouldn't exist, or at best be unreliable as w is not in the image of
> > Embedding(W,i).
> 
> Unfortunately, the same is true for Alexander's proposal - a generic  
> element
> in the base group does not lie in the image of any embedding.
> 
> I do not see any "clean" way of getting the components, either.
> So I guess we have to think about adding suitable `Projection'  
> methods as well.
> 
> > Am I taking a chance with using it?  Or does it indeed always give
> > the right thing?
> 
> I wouldn't rely on it (although in your case, it seems to work).  
> Probably depends on the kind of group the bottom group is, though.
> 
> > Perhaps you could suggest the right part of the GAP code I should look
> > at to create a version, as it probably would be worth my time to get a
> > reliable, efficient version of this as I will need to use it thousands
> > (if not millions) of times in a test for fixed point free actions of
> > certain groups I'm trying to construct.
> 
> This depend on the kinds of groups do you use to construct for the  
> wreath products (perm groups, pc groups, anything else). Maybe you  
> can send sample input to support at gap-system.org, which might be a  
> more appropriate place for such a technical discussion.
> 
> Cheers,
> 
> Burkhard.
> 
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-- 
Steve Linton	School of Computer Science  &
      Centre for Interdisciplinary Research in Computational Algebra
	     University of St Andrews 	 Tel   +44 (1334) 463269
http://www.cs.st-and.ac.uk/~sal 	 Fax   +44 (1334) 463278