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

[Max Horn]

Dear Petr,

I only saw your second email after writing my reply.

On 17.07.2014, at 14:37, Petr Savicky <savicky at cs.cas.cz> wrote:

> On Thu, Jul 17, 2014 at 08:54:41AM +0200, Benjamin Sambale wrote:
>> Dear Petr,
>> 
>> I don't see how your first question is related to the group G. If you 
>> want ALL extensions of A with a group of order 2, you could use 
>> CyclicExtensions(A,2) from the GrpConst package. However, if A is small, 
>> it is much faster to run through the groups of order 2|A| in the small 
>> groups library and check which groups have maximal subgroups isomorphic 
>> to A (i.e. the same GroupID).
> 
> Thank you for your reply. The extensions are considered as subgroups
> of G and the embedding is important, not only the isomorphism type.
> Consider the groups
> 
>  G := Group( [ (1,9)(2,10)(3,11)(4,12)(5,13)(6,14)(7,15)(8,16),
>       (1,5)(2,6)(3,7)(4,8), (1,3)(2,4), (1,2) ] );
> 
>  A := Group( [ (3,4)(5,8,6,7)(11,12)(13,14), (3,4),
>       (1,3)(2,4)(7,8)(11,12)(15,16), (3,4)(7,8)(9,12,10,11)(13,15,14,16),
>       (13,14)(15,16) ] );
> 
> Group G has order 32768, group A has order 256 and is isomorphic
> to SmallGroup(256, 27634).

Applying the two algorithms to this pair (G,A) takes approximately the same time for me -- in both case just a dozen milliseconds or so.

> 
> There are 19 extensions B of A inside G with the quotient
> group B/A = C_2.

Indeed.

> Some of these extensions are conjugate. For example, the groups
> B[16], ..., B[19] belong to the same conjugacy class.

Indeed. If you only want conjugacy classes of the extensions, a first step would be to adapt "my" algorithm by taking only one involution from each conjugacy class of the group H = N_G(A)/A. However, there can still be further groups which are conjugate in G (the classes of involutions in H may fuse when lifted to G etc.). 


[...]

> Is there a way, perhaps not a very efficient one, how to identify
> the groups of order 512 in the library?

This can be done with the anupq package, and the following function (taken from the SCSCP package manual, chapter 7):

IdGroup512ByCode := function( code )
  local G, F, H;
  G := PcGroupCode( code, 512 );
  F := PqStandardPresentation( G );
  H := PcGroupFpGroup( F );
  return IdStandardPresented512Group( H );
end;


However, I am somewhat skeptical whether going this way is a good idea... but then we don't know much about the problem you want to solve (in particular: The groups you are interested in), so...


Cheers,
Max