[GAP Forum] elements of wreath products?

[Burkhard Höfling]

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.