Dear forum, Dear William, > > I see how to use the Projection function to compute projections of a > product group. Is it also possible to compute projections of a > subgroup of a product group? Apparently this is not done by simply > invoking the Projection function, as the following example shows: > > G := Group([(1,2,3,4),(1,2)]);; > H := Subgroup(G, [(1,2)]); > K := Subgroup(G, [(1,2,3,4)]); > G1xG2 := DirectProduct(G, G); > emb1 := Embedding(G1xG2, 1);; > emb2 := Embedding(G1xG2, 2);; > h := List(GeneratorsOfGroup(H), i->Image(emb1,i)); > k := List(GeneratorsOfGroup(K), i->Image(emb2,i)); > HxK := Group(Concatenation(h, k),());; > > Now Projection(G1xG2,1) and Projection(G1xG2,2) work as expected, but > Projection(HxK,1) gives: > > Error, no method found! For debugging hints type ?Recovery from NoMethodFound
Projection returns a stored projection function of a direct or semidirect product. It thus applies only to the full product. However, as it is a homomorphism, it of course can be applied to any subgroup of the product, so you could take for example proj:=Projection(G1xG2,1); Image(proj,HxK); If you want to restrict the projection to the subgroup (primarily of relevance for computing pre-images) you could use rest:=RestrictedMapping(proj,HxK); > Of course, this is just a toy example -- we already know the > projections of HxK! -- but I would like to somehow compute the > projections of more general subgroups, as in > > intsub:=IntermediateSubgroups(G1xG2, HxK); > S:=intsub.subgroups[1]; > Projection(S, 1); # returns an error, as above. Again here I would simply use the same projection `proj' as long as it comes to computing images, or use the restricted mapping. > P.S. An unrelated, less pressing question, which I've been unable to > find in the manual or forum archives: Is there is command for changing > the current working directory in GAP? In the IO package there is IO_chdir if you are working under Unix. Best, Alexander (Hulpke) -- Colorado State University, Department of Mathematics, Weber Building, 1874 Campus Delivery, Fort Collins, CO 80523-1874, USA email: hul...@math.colostate.edu, Phone: ++1-970-4914288 http://www.math.colostate.edu/~hulpke _______________________________________________ Forum mailing list Forum@mail.gap-system.org http://mail.gap-system.org/mailman/listinfo/forum
