[GAP Forum] projections of subgroups of product groups

[William DeMeo]

Perfect.  Thank you!  -wjd

On Tue, Mar 29, 2011 at 2:27 PM, Alexander Hulpke <ahulpke at gmail.com> wrote:
>  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
>
>
>
>