> < ^ Date: Mon, 28 Jul 2003 15:58:45 +0200
< ^ From: Werner Nickel <nickel@mathematik.tu-darmstadt.de >
< ^ Subject: Re: Polycyclic group

Dear GAP Forum,
dear Primoz,

I have the following question: Given a (finite) pc-group G of rank r, is it
possible to construct (with the help of GAP) 'the largest' group H of the
same rank, such that H/Z(H) is isomorphic to G? Here 'the largest' means
that every other group of rank r with the above property is a homomorphic
image of H.

Let G be given by a finite presentation F/R where F is the free group
of rank r and R is the normal closure of the relations. Then F/[R,F]
is the largest central downward extension of G which is a factor group
of F. Clearly, R/[R,F] is a subgroup of the centre of F/[R,F].
However, the centre of F/[R,F] can be strictly larger. The easiest
case where this happens is G=Z/2Z, F=Z, R=2Z. Here, F/[R,F] = Z.

I have private code based on the polycyclic package that constructs
for a finite pc-group given by a finite presentation F/R the
polycyclic group F/[R,F]. I have sent the code to Primoz in a private
email, it is available from me on request.

The package 'polycyclic' is available at

http://www.mathematik.tu-darmstadt.de/~nickel/polycyclic

Regards,
Werner Nickel.

```--
Dr (AUS) Werner Nickel         Mathematics with Computer Science
Room:  S2 15/212                    Fachbereich Mathematik, AG 2
Tel:   +49 6151 163487                              TU Darmstadt
Fax:   +49 6151 166535                       Schlossgartenstr. 7
Email: nickel@mathematik.tu-darmstadt.de       D-64289 Darmstadt
--
```

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