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: [EMAIL PROTECTED], Phone: ++1-970-4914288
http://www.math.colostate.edu/~hulpke
_______________________________________________
Forum mailing list
[email protected]
http://mail.gap-system.org/mailman/listinfo/forum
