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

[Werner Nickel]

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
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   Email: nickel at mathematik.tu-darmstadt.de       D-64289 Darmstadt
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