[GAP Forum] p-quotient of an infinite matrix group
Werner Nickel
nickel at mathematik.tu-darmstadt.de
Fri Apr 30 12:31:07 BST 2004
Dear Gap Forum, dear Marco,
> 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 don't have an out-of-the box solution for Marco's question. I would
like to make the following suggestions:
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.
As far as I could see the resulting matrix group is not easy to
analyze with the standard GAP functions. Here is a strategy for an
analysis by hand:
Take the (images of the) conjugates of s by the powers of r. This
gives 7 upper triangular matrices. The subgroup generated by their
commutators contains matrices of the shape
[ 1 * * * * * * * ]
[ 1 0 0 0 0 0 0 ]
[ 1 0 0 0 0 0 ]
....
[ 1 ]
generating a subgroup of F^7. It is easy to determine a basis (over
the prime field) for the subgroup. After that it is easy to specify
the action of the conjugates of s on the commutator subgroup and
finally to add the action of r on the group generated by the
conjugates of s. After all this is done one ends up with a
pc-presentation for the image group.
If one chooses F to be GF(7) then most of the structure of the group
disappears because the only 7-th root of unity in characteristic 7 is
1.
Hope this helps,
Werner Nickel
--
Dr (AUS) Werner Nickel Mathematics with Computer Science
Room: S2 15/212 Fachbereich Mathematik, AG 2
Tel: +49 6151 163487 TU Darmstadt
Fax: +49 6151 166535 Schlossgartenstr. 7
Email: nickel at mathematik.tu-darmstadt.de D-64289 Darmstadt
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