[GAP Forum] projections of subgroups of product groups

[William DeMeo]

P.P.S.  I see why Projection(S,1) shouldn't work for an arbitrary
subgroup S of a direct product, since projection is not generally a
group homomorphism, and projections of S will not generally be
subgroups of S.  Still, we should be able to compute projections, even
when S is not directly factorable, right?

On Mon, Mar 28, 2011 at 11:45 PM, William DeMeo <williamdemeo at gmail.com> wrote:
> Dear Forum,
>
> This is probably a frequently asked and/or dumb question.  If so, I
> apologize in advance, but I couldn't find the answer in the forum
> archives.
>
> 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
> Error, no 1st choice method found for `Projection' on 2 arguments called from
> <function>( <arguments> ) called from read-eval-loop
> Entering break read-eval-print loop ...
> you can 'quit;' to quit to outer loop, or
> you can 'return;' to continue
>
> 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.
>
> Thank you for any suggestions you can offer.
>
> -William
>
> 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?
>