> < ^ Date: Thu, 08 May 2003 17:01:38 +0200
^ From: Enrique Artal Bartolo <artal@unizar.es >
^ Subject: Rings, monoids and subgroups

Dear GAP Forum:
I am interested in computing finite representations of the pure braid group
starting from Burau and Gassner representations.

For example, the Burau representation maps into the invertible matrices over
the ring of Laurent polynomials with integer coefficients,which is the group
ring of Z. A simple idea is to replace Z by Z mod n and either replace Z by a
finite cyclic group or mod out by an ideal, e.g. (t-1)^2.

I have tried the first approach by constructing the group ring R of Z/mZ with
coefficients in Z/nZ, which is straightforward in GAP. Then I consider the
ring MR of square matrices with coefficients in R of size, say 4.

It is possible to define a subset H of MR which is the image of a
representation of the pure braid group and it is a subgroup of the invertible
matrices. The problem is that GAP only recognize a monoid structure on H;
this monoid is finitely generated and the inverses of the generators can be
computed by hand. Is there an easy way to help GAP recognize the group
structure of H?

Thanks, Enrique ARTAL.

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