[GAP Forum] extensions of subgroups of a finite 2-group
Max Horn
max at quendi.de
Thu Jul 17 17:09:05 BST 2014
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
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