[GAP Forum] Marco's question: p-quotient of an infinite matrix group

[Robert Eckert]

On Fri, 30 Apr 2004, Werner Nickel wrote:
...> > let G be an infinite matrix group, like in the example below...
> > 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.
Computationally cleanest would be GF(64) with defining polynomial
x^6 = x^5 + x^4 + x^3 + x^2 + x + 1 = x^(-1)
Elements could be written with x^0 to x^5 in bits 1 to 6 of a byte leaving
bits 0 and 7 clear; then "times/divide x" is a one-position rotate,
followed, if bit 0 or 7 is left set, by setting the other and then
complementing the whole byte.
This group is acting on the dim-8 vectors (really dim-7 as the first
coefficient is always fixed, but I assume this is not always the case in
Marco's groups).  Surely GAP give an fp-presentation of SL(8,64); to find
the appropriate subgroup, note that if the defining polynomial for GF(64)
is other than the above, the actual seventh-root of one must be used
instead of just "x".
Bob Eckert (Wayne State)