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 <[EMAIL PROTECTED]> 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 [EMAIL PROTECTED], 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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