[GAP Forum] How to find consistent polycyclic presentation

[Petr Savicky]

On Wed, Aug 26, 2015 at 12:07:45PM +0530, Vipul Kakkar wrote:
> Dear Members
> 
> How can I find consistent polycyclic presentation form
> (1) the group that I chose from GAP library and
> (2) the group that I define by FreeGroup/Relation.
> 
> For example, how to find consistent polycyclic presentation
> 
> for g:=SmallGroup(81,5); and
> 
> for f:=FreeGroup(2); g:=f/[f.1^4, f.2^2, (f.1*f.2)^2];

The polycyclic presentation can be displayed using Polycyclic package.
In the first case one can use

  g := SmallGroup(81,5);
  g1 := ImagesSource(IsomorphismPcpGroup(g));
  PrintPcpPresentation(g1);

  g1^3 = g3
  g2^3 = id 
  g3^3 = g4
  g4^3 = id 

In the second case, the group has to be first converted to a polycyclic
representation of pc type and then displayed as above

  f:=FreeGroup(2); g:=f/[f.1^4, f.2^2, (f.1*f.2)^2];
  h := ImagesSource(IsomorphismPcGroup(g));
  g2 := ImagesSource(IsomorphismPcpGroup(h));
  PrintPcpPresentation(g2);

  g1^2 = g3
  g2^2 = id 
  g3^2 = id 
  g2 ^ g1 = g2 * g3

Best regards,
Petr Savicky.