[GAP Forum] elements of wreath products?

[R. Keith Dennis]

Dear Alexander,

thanks very much for the reply.

> > I'm sure this is trivial, but the reference manual seems unclear on
> > this, and the "obvious" thing doesn't seem to work.  If one looks at a
> > wreath product W:=WreathProduct(G,P), and thinks of an element as
> > being represented as a product bp (or the other way around) where b is
> > in the base group and p is in the permuation group P (b =
> > (b_1,...,b_n), the simplest version:  G^n semi P, where n is the order
> > of P), how does one find the coordinates of an element w in W?  I.e.,
> > the b_i.  Presumably this should be given by Image(Projection(W,i),w)
> > for i between 1 and n, however that leads to an error message.  It
> > seems I'm missing something.  What?
>
> The problem stems from the fact that the only mappings defined for  
> wreath products are group homomorphisms -- projection onto the G- 

Ah, a reasonable convention.  But sometimes one needs other maps as well.

> components is not a group homomorphism and therefore does not exist.
>
> 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).

Am I taking a chance with using it?  Or does it indeed always give
the right thing?

> Admittedly this looks a bit contorted. I'd be happy to entertain the  
> introduction of a special operation to decompose elements of a wreath  
> product if anybody needs to do this kind of decomposition more often  
> or time-critical.

That in fact seems like a reasonable thing to do at some point.  I
haven't checked, but perhaps one would like the same sort of
non-homomorphism in the case of a non-trivial semi-direct product (or
maybe it's already there?).

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.

Thanks again for your help.

Best regards,

Keith