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Dear David Cruickshank,

You asked about group rings and matrices defined over them in GAP.

In GAP4 it is possible to construct the group ring of a group:

gap> G:= AbelianGroup( [1,2,3,4,5]);

<pc group of size 120 with 5 generators>

gap> ZG:= GroupRing( Integers, G );

<free left module over Integers, and ring-with-one, with 8 generators>

Then it is also possible to define matrices over the group ring

and calculate with them:

gap> B:=Basis(ZG);

CanonicalBasis( <free left module over Integers, and ring-with-one, with

8 generators> ) gap> m:= [ [ B[1], B[10] ],[ B[40], B[100] ] ]; [ [ 1*<identity> of ..., 1*f1*f5 ], [ 1*f1*f2^2*f5, 1*f2*f3*f5^4 ] ] gap> DeterminantMat(m); -1*f2^2*f5^2+1*f2*f3*f5^4

However, as yet GAP does not contain any code for calculating

Groebner bases.

I hope this answer is of help to you. If you have any

further questions, please ask.

Best wishes,

Willem de Graaf

School of Mathematical and Computational Sciences University of St Andrews North Haugh Tel: +44 1334 463273 St Andrews Email: wdg@dcs.st-and.ac.uk Fife KY16 9SS Scotland

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