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

Werner Nickel nickel at mathematik.tu-darmstadt.de
Mon May 3 08:17:06 BST 2004

Dear Robert,

> > 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.

you correctly point out that arithmetic in GF(64) is efficient if one
chooses an adequate representation for the elements of the field.
However, efficiency of the field arithmetic is not the main difficulty
in Marco's question.  The main point is to find a homomorphism into
a finite (p-) group.  Once you have defined a homomorphism into a
matrix group over a suitable finite field, the speed of the underlying
arithmetic plays a role but most likely not a prominent one.  By the
way, GAP implements elements of GF(64) (and all finite fields of
small order) by storing the power of an element with respect to fixed
primitive element of the field.  Multiplication reduces to addition
modulo the order of the mutliplicative group.  Addition is essentially
encoded by an addition table.

> 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); 

A finite presentation for SL(8,64) does not help much in finding a
finite presentation for Marco's group because it is a difficult to
derive a presentation for a subgroup from one of the whole group, in
particular if the index of the subgroup is big as it is in this case.

With kind regards,
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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