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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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