# [GAP Forum] Subgroup stable under transposition

Alexander Hulpke ahulpke at gmail.com
Wed Oct 8 05:02:09 BST 2008

```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: hulpke at math.colostate.edu, Phone: ++1-970-4914288
http://www.math.colostate.edu/~hulpke

```