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 _______________________________________________ Forum mailing list [email protected] http://mail.gap-system.org/mailman/listinfo/forum
