Dear Dan Lanke, Dear Forum,
Let A be a finite (abelian) group. How do I list all subgroups of A
x A that are stable under the transposition (a,b) --> (b,a)?
I know how to list all subgroup, but I don't know how to take care
of the transposition condition.
You can use the routine to calculate invariant subgroups, constructing
the flip as automorphism of AxA.
For example:
gap> A:=TransitiveGroup(8,2); # some abelian group
4[x]2
gap> D:=DirectProduct(A,A);
Group([ (1,2,3,8)(4,5,6,7), (1,5)(2,6)(3,7)(4,8), (9,10,11,16)
(12,13,14,15),
(9,13)(10,14)(11,15)(12,16) ])
Now construct the flipper automorphism of D:
gap> hom:=GroupHomomorphismByImages(D,D,Concatenation(A1gens,A2gens),
> Concatenation(A2gens,A1gens));
[ (1,2,3,8)(4,5,6,7), (1,5)(2,6)(3,7)(4,8), (9,10,11,16)(12,13,14,15),
(9,13)(10,14)(11,15)(12,16) ] ->
[ (9,10,11,16)(12,13,14,15), (9,13)(10,14)(11,15)(12,16), (1,2,3,8)
(4,5,6,7),
(1,5)(2,6)(3,7)(4,8) ]
gap> IsBijective(hom);
true
Now construct subgroups invariant under hom:
s:=SubgroupsSolvableGroup(D,rec(actions:=[hom]));
I hope this is of help,
Alexander Hulpke
-- Colorado State University, Department of Mathematics,
Weber Building, 1874 Campus Delivery, Fort Collins, CO 80523-1874, USA
email: [EMAIL PROTECTED], Phone: ++1-970-4914288
http://www.math.colostate.edu/~hulpke
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