[GAP Forum] Heisenberg group from the Polycyclic package

[Bettina Eick]

Dear Ha T. Lam,

> I'm doing some computations with Heisenberg group and I'm using the method
> HeisenbergPcpGroup(n) from the Polycyclic package. According to the manual,
> I should get a Heisenberg group of 2n generators with Hirsch length 3n. I
> do:
>
> gap> HeisenbergPcpGroup(3);
> Pcp-group with orders [ 0, 0, 0, 0, 0, 0, 0 ]
>
> The Hirsch length here is 7, not 9. Is this a mistake in the manual? Can
> you give me some reference of how these groups are generated by this
> package?

It is indeed a mistake in the manual. The function HeisenbergPcpGroup(m)
returns the Heisenberg group on 2m generators; this has Hirsch length 
2m+1. To see some more details on how these groups are generated, you 
can do

gap> Print(HeisenbergPcpGroup);
function ( m )
     local  FLT, i;
     FLT := FromTheLeftCollector( 2 * m + 1 );
     for i  in [ 1 .. m ]  do
         SetConjugate( FLT, m + i, i, [ m + i, 1, 2 * m + 1, 1 ] );
     od;
     UpdatePolycyclicCollector( FLT );
     return PcpGroupByCollectorNC( FLT );
end

Thus HeisenbergPcpGroup(m) returns a group on 2m+1 polycyclic generators
g_1, ..., g_{2m+1} with conjugate relations

   g_{m+i}^g_i = g_{m+i} g_{2m+1} for 1 <= i <= m

Best wishes,

Bettina