[GAP Forum] elements of wreath products?

[Alexander Hulpke]

On Sep 11, 2007, at 2:02 PM, Steve Linton wrote:

> It seems to me we need a slightly more general function than  
> Projection.
>
> Maybe something like Components(W ,w) returning a list of n+1  
> elements of
> suitable subgroups.

Something like this -- also for semidirect products -- looks the best  
to me and I will implement such functions.
>
> Alternatively, and perhaps cleaner, one could provide a way to get  
> at the
> subgroup of W which is a direct product with its Projections and  
> Embeddings.

I communicated privately with Keith Dennis -- his application is time  
sensitive which makes it worth not having to go through multiple  
mappings.

Best,

     Alexander


>
> 	Steve
>
> On Tue, 11 Sep 2007 14:50:04 +0200
> Burkhard Höfling <burkhard at hoefling.name> wrote:
>
>> Dear Keith, dear all,
>>
>>>> What exists for a wreath product as described is
>>>> Projection(W);  # no index!
>>>> which is the projection onto P,
>>>> Embedding(W,i) # i=1..n
>>>> the homomorphism G->W giving the i-th copy of G and
>>>> Embedding(W,n+1)
>>>> giving the complement P to G^n.
>>>>
>>>> To get the i-th component of an element x thus one needs to split
>>>> off-
>>>> the p-part first and then use the pre-image under a suitable
>>>> embedding:
>>>>
>>>> PreImagesRepresentative(Embedding(W,i),x/Image(Embedding(W,n 
>>>> +1),Image
>>>> (Projection(W),x)));
>>>
>>> Laurent Bartholdi had already suggested that I try
>>>
>>>   PreImagesRepresentative(Embedding(W,i),w)
>>>
>>> which seems to work.  However, the manual seems to suggest that this
>>> shouldn't exist, or at best be unreliable as w is not in the  
>>> image of
>>> Embedding(W,i).
>>
>> Unfortunately, the same is true for Alexander's proposal - a generic
>> element
>> in the base group does not lie in the image of any embedding.
>>
>> I do not see any "clean" way of getting the components, either.
>> So I guess we have to think about adding suitable `Projection'
>> methods as well.
>>
>>> Am I taking a chance with using it?  Or does it indeed always give
>>> the right thing?
>>
>> I wouldn't rely on it (although in your case, it seems to work).
>> Probably depends on the kind of group the bottom group is, though.
>>
>>> Perhaps you could suggest the right part of the GAP code I should  
>>> look
>>> at to create a version, as it probably would be worth my time to  
>>> get a
>>> reliable, efficient version of this as I will need to use it  
>>> thousands
>>> (if not millions) of times in a test for fixed point free actions of
>>> certain groups I'm trying to construct.
>>
>> This depend on the kinds of groups do you use to construct for the
>> wreath products (perm groups, pc groups, anything else). Maybe you
>> can send sample input to support at gap-system.org, which might be a
>> more appropriate place for such a technical discussion.
>>
>> Cheers,
>>
>> Burkhard.
>>
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>
>
> -- 
> Steve Linton	School of Computer Science  &
>       Centre for Interdisciplinary Research in Computational Algebra
> 	     University of St Andrews 	 Tel   +44 (1334) 463269
> http://www.cs.st-and.ac.uk/~sal 	 Fax   +44 (1334) 463278
>
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