[GAP Forum] Metabelian

[Max Horn]

Am 18.12.2011 um 17:00 schrieb Mr. Sorouhesh:

> Dear Froum,
> 
> Is there any way in GAP that one can check a finitely presented group is a
> metabelian?

Dear Mr. Sorouhesh,

as you may know, in general it is not even possible to decide for a given finitely presented group whether it is trivial; this then extends to verifying many other properties, including whether a group is metabelian. However, in some specific cases, more is possible, and this may be the case for you, too -- but without more details about your specific problem, I can't tell. 

Anyway, here is a rough strategy as to what you can do: Given the finite presentation G=F/R, start by computing a maximal polycyclic quotient. This can be done with my not yet released GAP package pcql (feel free to contact me in private if you would like to compute specific quotients). 

This yields a quotient H of G which is polycyclic. If this quotient is not metabelian (i.e. H'' is not trivial), then you can stop, as G is not metabelian either. So assume H is metabelian (not that for me, abelian groups are also metabelian). All we know is that it is a quotient of G, and we would like to verify that G and H are actually isomorphic. If G is actually polycyclic, then in Sims' book "Computation with finitely presented groups", section 11.8 there is a strategy on how to do that, given the polycyclic quotient H. (See also <http://www.gap-system.org/Faq/Computing/computing10.html> for a concrete example where this is being applied).

If, however G is *not* polycyclic, then this can in general not be decided (this is similar to deciding whether a subgroup of a finitely presented group has finite index -- if it has, you can compute it and thus have a proof; but if you cannot compute the index, then you do not know whether you simply did not try hard enough, of whether the index is infinite).

Of course, in your specific case, it may still be possible to resolve the problem completely; again, it depends on the specific finite presentation.

One last remark: Above I only talked about polycyclic metabelian quotients. Of course in general finitely presented metabelian groups need not be polycyclic (Baumslag and other constructed examples for this). With my package "pcql", you can still compute a maximal metabelian quotient, even if it is not polycyclic, and even compute normalforms of group elements in it or obtain a power-conjugator presentation for the group. With luck, this may be sufficient to resolve your problem, but the general strategy described by Sims cannot be applied directly in this case.


Best regards,
Max
-- 
Dr. Max Horn
AG Algebra und Diskrete Mathematik
Institut Computational Mathematics
TU Braunschweig
Pockelsstrasse 14, D-38106 Braunschweig
Tel: (+49) 531 391-7526
Fax: (+49) 531 391-7414
Web: http://www.icm.tu-bs.de/~mhorn/
Email: mhorn at tu-bs.de