> < ^ Date: Sat, 27 Feb 1999 16:47:34 +0000
> < ^ From: Willem de Graaf <degraaf@math.uu.nl >
< ^ Subject: Re: Beginner's Problems

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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