[GAP Forum] projections of subgroups of product groups

[Alexander Hulpke]

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: hulpke at math.colostate.edu, Phone: ++1-970-4914288
http://www.math.colostate.edu/~hulpke