Dear GAP Forum,
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
-- 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: firstname.lastname@example.org D-64289 Darmstadt --