> < ^ Date: Thu, 18 May 2000 09:41:00 +0200 (MET DST)
> < ^ From: Jan Draisma <Jan.Draisma@unibas.ch >
< ^ Subject: Re: A problem with "FreeMagmaRing"

Dear GAP Forum,

In answer to Ignat Soroko's question: the elements g1 and g2 are not
invertible, if GF(3) in the example from the manual is replaced by
GF(2). A brute force way to see this is:

```gap> Length(AsSet(List(AsList(rm),x->x*g1)));
16
gap> Size(rm);
256
gap> Length(AsSet(List(AsList(rm),x->x*g2)));
4
```

Regards,

Jan Draisma

```In answer to
> Dear GAP Forum,
>
> I was trying to execute example from the Reference Manual to GAP 4.2
> concerning group rings on page 567. For the field GF(3) all is OK, but
> when I changed to GF(2) an error occured:
>
> gap> d8:=DihedralGroup(8);
> <pc group of size 8 with 3 generators>
> gap> rm:=FreeMagmaRing(GF(2),d8);
> <algebra-with-one over GF(2), with 3 generators>
> gap> emb:=Embedding(d8,rm);
> <mapping: Group( [ f1, f2, f3 ] ) -> AlgebraWithOne( GF(2), ... ) >
> gap> gens:=List(GeneratorsOfGroup(d8),x->x^emb);
> [ (Z(2)^0)*f1, (Z(2)^0)*f2, (Z(2)^0)*f3 ]
> gap> x1:=gens[1]+gens[2];;
> gap> x2:=(gens[1]-gens[2])*gens[3];;
> gap> x3:=gens[1]*gens[2]*(One(rm)-gens[3]);;
> gap> g1:=x1-x2+x3;;
> gap> g2:=x1+x2;;
> gap> q8:=Group(g1,g2);;
> Error usage: Group(<gen>,...), Group(<gens>), Group(<gens>,<id>) at
> Error( "usage: Group(<gen>,...), Group(<gens>), Group(<gens>,<id>)" );
> <function>( <arguments> ) called from read-eval-loop
> Entering break read-eval-print loop, you can 'quit;' to quit to outer loop,