[GAP Forum] extensions of subgroups of a finite 2-group
Max Horn
max at quendi.de
Thu Jul 17 15:29:18 BST 2014
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
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