There are two problems with your code.
The first is that LatticeSubgroups is not the set of subgroups but a
different kind of structure. If you want to access all the subgroups,
you should use the function Subgroups. This requires the SONATA
package or some other equivalent package -- you can load that using
LoadPackage("sonata");.
The problem is that GAP doesn't understand "HK" to be the product of H
and K but rather thinks that that is a new variable.
Copy and paste this code in the GAP interface (or put it in a file and
read the file through GAP):
LoadPackage("sonata");
ProductOfSubsets := function(A,B)
local a,b,L;
L := [];
for a in Set(A) do
for b in Set(B) do
Add(L,Product([a,b]));
od;
od;
return Set(L);
end;;
PermutingSubsets := function(H,K)
return(ProductOfSubsets(H,K) = ProductOfSubsets(K,H));
end;;
NumberOfPermutingSubgroupPairs := function(G)
local S,F;
S := Subgroups(G);
F := Filtered(Cartesian(S,S), x -> PermutingSubsets(x[1],x[2]));;
return Length(F);
end;;
end of code
Now you can do:
NumberOfPermutingSubgroupPairs(AlternatingGroup(4));
The answer in this case seems to be 64, for what it's worth.
* Quoting Stefanos Dalamaidhs who at 2012-03-08 04:43:40+0000 (Thu) wrote
> Dear Forum,
>
> I would like to write a short code to do the following task:for any two
> subgroups of a group G, check whether their product is a subgroup of G and
> count their number. A failed attempt was:
>
> G:=AlternatingGroup(4);
> Alt( [ 1 .. 4 ] )
>
> L:=LatticeSubgroups(G);
>
> count:=0;
>
> for H in L do;
> for K in L do;
> if IsSubgroup(G,HK) then count:=count+1;
>
> fi;
> od;
> od;
> 0
> Error, no method found! For debugging hints type ?Recovery from
> NoMethodFound
> Error, no 1st choice method found for `Enumerator' on 1 arguments
>
> I would be obliged for any help.
>
> Thanks,
> J.
> _______________________________________________
> Forum mailing list
> Forum@mail.gap-system.org
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