[GAP Forum] projections of subgroups of product groups

[William DeMeo]

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?