[GAP Forum] p-quotient of an infinite matrix group
Bjoern Assmann
bjoerna77 at gmx.de
Tue May 4 11:23:19 BST 2004
Dear GAP Forum, dear Marco,
on Thu, 22 Apr 2004 Marco Costantini wrote
> let G be an infinite matrix group, like in the example below.
> I'd like to study a p-quotient (or a nilpotent-, solvable-,
> polycyclic- quotient) of > G (I mean a quotient of the
> form G / PCentralSeries(G,7)[n]). The obstacles are that the quotient
> methods requires a finitely presented group, and the conversion from
> matrix groups to finitely presented groups
> is available only for finite groups.
>
> Are there any suggestion?
>
> r :=
> [ [ 1, 0, 0, 0, 0, 0, 0, 0 ], [ 0, 0, 1, 0, 0, 0, 0, 0 ],
> [ 0, 0, 0, 1, 0, 0, 0, 0 ], [ 0, 0, 0, 0, 1, 0, 0, 0 ],
> [ 0, 0, 0, 0, 0, 1, 0, 0 ], [ 0, 0, 0, 0, 0, 0, 1, 0 ],
> [ 0, 0, 0, 0, 0, 0, 0, 1 ], [ 0, 1, 0, 0, 0, 0, 0, 0 ] ];
> s :=
> [ [ 1, 0, 0, 0, 0, 0, 0, 1 ], [ 0, E(7)^6, 0, 0, 0, 0, 0, 0 ],
> [ 0, 0, E(7), 0, 0, 0, 0, 0 ], [ 0, 0, 0, 1, 0, 0, 0, 0 ],
> [ 0, 0, 0, 0, 1, 0, 0, 0 ], [ 0, 0, 0, 0, 0, 1, 0, 0 ],
> [ 0, 0, 0, 0, 0, 0, 1, 0 ], [ 0, 0, 0, 0, 0, 0, 0, 1 ] ];
>
> G:=Group( [ r, s ] );
I think that I can solve Marco's problem using the methods of the Polenta
package. This package contains an algorithm to determine a polycyclic
presentation for a polycyclic rational matrix group. A variation of the
method applies to Marco's group. I used this variation to compute a
polycyclic presentation on 70 generators for the considered matrix group
in GL(8, Q(E(7))). The factors of its polycyclic series are
[ 7, 7, 7, 7, 7, 7, 7, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2,
2, 2, 2, 2, 2, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
0, 0, 0, 0 ]
where 0 stands for an infinite factor. Using the polycyclic presentation
it is now straightforward to compute all kinds of factor groups for the
considered group (these methods are implemented in the Polycyclic
package). For example, the orders of the first 7 factors of the
7-central series of the group are:
49, 7, 7, 7, 7, 7, 49
The Polenta package is using a similar approach as suggested by Werner:
it first computes a polycyclic presentation for a finite image of the
group in GL( d, q ) for some suitable prime power q and then it computes
a polycyclic presentation for the kernel of this map.
The practicality of the method depends critically on the choosen prime
power q and also, in Marco's case, I needed to extend the method from
working over the rationals to working over an algebraic number field.
Hence the Polenta package does not solve Marco's problem without some
additional tricks. These tricks are available on
http://cayley.math.nat.tu-bs.de/software/assmann/marcosgroup.g
gap> K := PcpGroupOfMarcosExam();
Pcp-group with orders [ 7, 7, 7, 7, 7, 7, 7, 2, 2, 2, 2, 2, 2, 2, 2,
2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
0, 0, 0, 0, 0, 0, 0, 0, 0 ]
gap> time;
1027770
# this are approximately 17 minutes
Marco wrote
> I'd like to
> study a p-quotient (or a nilpotent-, solvable-, polycyclic- quotient) of
> G (I mean a quotient of the form G / PCentralSeries(G,7)[n]).
The function call PCentralSeriesSteps( K, 7, n) caculates the first n+1
Groups of the 7-central series (this function
written by Bettina is also contained in the file marcosgroup.g).
gap> ser := PCentralSeriesSteps(K,7,7);;
gap> Order( ser[1]/ser[2] );
49
gap> Order( ser[2]/ser[3] );
7
Maybe it suffices to study a quotient of K (see the ANUPQ Package).
Short Desription of the implemented method:
------------------------------------------
on Fri, 30 Apr 2004 Wernel Nickel wrote
> There is a map of rings from the 7-th cyclotomic integers into a
> finite field F containing a 7-th root of unity. The map is
> essentially specified by mapping E(7) to the 7-th root of unity in F.
> This map defines a map from G into the (invertible) matrices over F.
Following the suggestion from Werner I chosed a map from the 7-th
cyclotomic integers into the
Galois Field GF(2^3) (mapping E(7) to Z(2^3)).
This map gives rise to a map
phi : G -> GL(8,GF(2^3))
The images of r and s under phi are
phi(r) =
[ [ Z(2)^0, 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2) ],
[ 0*Z(2), 0*Z(2), Z(2)^0, 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2) ],
[ 0*Z(2), 0*Z(2), 0*Z(2), Z(2)^0, 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2) ],
[ 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2), Z(2)^0, 0*Z(2), 0*Z(2), 0*Z(2) ],
[ 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2), Z(2)^0, 0*Z(2), 0*Z(2) ],
[ 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2), Z(2)^0, 0*Z(2) ],
[ 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2), Z(2)^0 ],
[ 0*Z(2), Z(2)^0, 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2) ] ];
phi(s) =
[ [ Z(2)^0, 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2), Z(2)^0 ],
[ 0*Z(2), Z(2^3)^6, 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2) ],
[ 0*Z(2), 0*Z(2), Z(2^3), 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2) ],
[ 0*Z(2), 0*Z(2), 0*Z(2), Z(2)^0, 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2) ],
[ 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2), Z(2)^0, 0*Z(2), 0*Z(2), 0*Z(2) ],
[ 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2), Z(2)^0, 0*Z(2), 0*Z(2) ],
[ 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2), Z(2)^0, 0*Z(2) ],
[ 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2), Z(2)^0 ] ];
With the function "PcpGroupByMatGroup"
you can calculate a polycyclic presentation of phi(G).
Using the relations of this presenation you can calculate normal
subgroup generators k_1,...,k_n for the kernel of phi.
The matrices k_i are of the shape
1 * * * * * * *
0 1 0 0 0 0 0 0
...............
0 0 0 0 0 0 1 0
0 0 0 0 0 0 0 1
where the entries "*" are in the 7-th cyclotomic integers I.
Thus the kernel of phi is
isomorphic to a subgroup of the lattice I^7 and therefore polycyclic.
It follows that the hole group G is polycyclic.
Now it is easy to calculate a basis for the lattice, which is
isomorphic to <k_1,...,k_n>, and to
close this basis under the conjuation action of G.
Using the homomorpism from G onto the polycyclic presentation of
phi(G) and the basis for the kernel of phi, you can calculate a
presentation for the whole group G.
Remark 1
--------
In principal you could also represent G as 48 * 48 matrix group "G_big"
over the rationals. (Every entry of r and s would become a 6*6 matrix
and E(7) would be represented as
[ [ 0, 1, 0, 0, 0, 0 ],
[ 0, 0, 1, 0, 0, 0 ],
[ 0, 0, 0, 1, 0, 0 ],
[ 0, 0, 0, 0, 1, 0 ],
[ 0, 0, 0, 0, 0, 1 ],
[ -1, -1, -1, -1, -1, -1 ] ])
Then you could apply directly PcpGroupByMatGroup to G_big.
Unfortunately the computation costs too much memory in this case.
To be more precise the arrising orbits of a finite image of G_big
become to long.
Remark 2
--------
As Werner Nickel pointet out to me yesterday, his suggestion for the
computation of a presentation by hand for a finite image of G
( see his e-mail to the GAP Forum
from Friday, 30 Apr 2004) is also valid for G.
Best wishes
Bjoern
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