[GAP Forum] elements of wreath products?

[Alexander Hulpke]

Dear Keith,

> 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- 
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)));

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.

Best,

     Alexander

-- 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