> < ^ Date: Thu, 17 Jul 1997 10:47:07 +0200
> < ^ From: Bettina Eick <beick@tu-bs.de >
< ^ Subject: Re: function 'Centre' for special Ag groups

Dear Gap Forum,

Burkhard H"ofling wrote:

I would like to point out the following problem with the function 'Centre'
for special Ag groups: sometimes, the maximum size of a field in GAP is not
large enough to compute the centre of even a small special Ag group with
the function 'Centre'. The group in the following example has order
216=3D2^3.3^3. It is, in fact, a central product of SL(2,3) with a cyclic
group of order 18.

g :=3D Group( [ [ E(4), 0 ], [ 0, -E(4) ] ],
   [ [ 1/2*E(108)^56-1/2*E(108)^83, 1/2*E(108)^56-1/2*E(108)^83 ],
     [ -1/2*E(108)^56-1/2*E(108)^83, 1/2*E(108)^56+1/2*E(108)^83 ] ] );
a :=3D AgGroup (g);
s :=3D SpecialAgGroup (a);
Centre (s);

results in the following error message:

Error, Z: <q> must be a prime power in [2..65536] at
return Z( p ^ d ) ^ i ... in

Note that this is not a grave problem, since Centralizer (s,s) yields the
desired result in virtually no time.

Unfortunately, it is a feature of the underlying algorithm that it
needs large finite fields. Since GAP can only deal with finite
fields of size < 2^16 at the moment, the algorithm will run in an
error if it needs larger fields.

However, the underlying algorithm SagGroupOps.Centre is often faster
than the generic method AgGroupOps.Centre and the above bug does not
turn up very often. This is the reason why the algorithm is still
used as SagGroupOps.Centre.

Best wishes, Bettina

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