# [GAP Forum] p-quotient of an infinite matrix group

Fri Apr 30 12:31:07 BST 2004

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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