[GAP Forum] extensions of subgroups of a finite 2-group

[Max Horn]

Dear Petr,

On 16.07.2014, at 16:39, Petr Savicky <savicky at cs.cas.cz> wrote:

> Dear GAP Forum:
> 
> Assume, G is a finite 2-group and A is its subgroup.
> The groups may be permutation groups or pc groups.
> I would like to construct all extensions B of A, such
> that [B:A] = 2.

I assume you meant "all extensions B *in G* of A...".


> 
> One way is to perform
> 
>    N := Normalizer(G, A);
>    R := RightTransversal(N, A);
>    L := [];
>    for elm in R do
>        if elm in A then
>            continue;
>        fi;
>        if elm^2 in A then
>            Add(L, ClosureGroup(A, elm));
>        fi;
>    od;
> 
> Is there a better way?

Yes, there is, at least asymptotically -- it will be slower for small examples, but faster for larger ones. Do this:

1. Compute the quotient H:=N_G(A)/A
2. Compute the conjugacy classes of involutions in H
3. For each involution in H, its preimage in N_G(A) resp. G
   is a group with the desired property, and this correspondence 
   is bijective.

Here is a direct implementation:

    N := Normalizer(G, A);
    hom := NaturalHomomorphismByNormalSubgroup(N, A);
    H := ImagesSource(hom);
    cc := ConjugacyClasses(H);
    L := [];
    for cl in cc do
        if Order(Representative(cl)) <> 2 then
            continue;
        fi;
        for elm in cl do
            elm := PreImagesRepresentative(hom, elm);
            Add(L, ClosureGroup(A, elm));
        od;
    od;


To test it, I took

 G:=GL(4,8);
 A1:=SylowSubgroup(G,2);   # |H| = 2401
 A2:=DerivedSubgroup(A1); # |H| = 1229312


For A1, with "your" method, it takes 0.9s on my system, and 1.1s with the conjugacy class method -- so it is slower there.

But for A2, the conjugacy class method finished in in 3.7s, whereas "your" method took 493s.


> 
> Another question is as follows. Let G be a 2-group
> and H and A its subgroups, such that the intersection
> of H and A is trivial. Is it possible to determine
> in GAP, whether there is a subgroup B of G, such
> that B contains A and is a complement of H in G?

I am not aware of a direct method. When H is normal, maybe using ComplementClassesRepresentatives would help a bit, by reducing the problem to conjugacy classes of complements.


All the best,
Max