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
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