[GAP Forum] Alternative Methods for Constructing Non-Virtually Nilpotent Polycyclic Groups in GAP

[Max Horn]

Dear Jonathan,

> On 19 Dec 2016, at 02:55, Jonathan Gryak <gryakj at gmail.com> wrote:
> 
> Hello Forum Members,
> The GAP package Polycyclic contains the function MaximalOrderByUnitsPcpGroup
> <http://www.gap-system.org/Manuals/pkg/polycyclic-2.11/doc/chap6_mj.html#X78CEF1F27ED8D7BB>
> that constructs a polycyclic group that is infinite and non-virtually
> nilpotent. These groups are of the form O(F) \rtimes U(F), where F is an
> algebraic number field and O(F) and U(F) are respectively the maximal order
> and unit group of F, as discussed in the chapter on polycyclic group
> in the Handbook
> of Computational Group Theory
> <https://www.crcpress.com/Handbook-of-Computational-Group-Theory/Holt-Eick-OBrien/p/book/9781584883722>
> .
> 
> Is there another method for constructing infinite, non-nilpotent polycyclic
> groups that doesn't rely on the split extension method above? And can this
> method be implemented in GAP, say via a finite presentation?

This is the primary purpose of the Polycyclic package you already are using: It allows you to work with arbitrary Polycyclic groups. You can do so either by constructing a suitable collector directly (as described in Chapter 3 of the Polycyclic manual), or else by defining a suitable finitely presented group, then using the function IsomorphismPcpGroupFromFpGroupWithPcPres (see chapter 5 of the Polycyclic manual).

Regards,
Max