[GAP Forum] Reduced Multiplication

[Stephen Linton]

Dear Sopsku,

What you need to be aware of here is that the relators of a finitely-presented group are NOT 
elements of the group. They are elements of the underlying free group. So after your commands below we see

gap> rel[1] in d4;
false
gap> rel[1] in FreeGroupOfFpGroup(d4);
true

One way of connecting them is to make the natural quotient homomorphism:

gap> phi := GroupHomomorphismByImages(f,d4,GeneratorsOfGroup(f),GeneratorsOfGroup(d4));
[ r, s ] -> [ r, s ]
gap> List(rel, x->Image(phi,x));
[ <identity ...>, <identity ...>, <identity ...> ]
gap> 

This is a bit confusing here because the generators of the free group print as "r" and "s" as 
do the generators of the finitely-presented group, but th global variables r and s hold the generators of the fp group.

	Steve

On 26 Aug 2013, at 03:54, Sopsku <rrburns at cox.net> wrote:

> Hi all,
> I have some small confusion about reduced multiplication that I would like
> to clear up. Suppose I have:
> 
> gap> d4:=DihedralGroup(IsFpGroup,4);;
> gap> r:=d4.1;;s:=d4.2;;
> gap> SetReducedMultiplication(d4);
> 
> Then 
> 
> gap> r^2;
> <identity ...>
> 
> OK - now suppose I look at 
> 
> gap> rel:=RelatorsOfFpGroup(d4);
> [ r^2, s^2, s^-1*r*s*r ]
> 
> rel does not reduce and I guess that makes sense as I would want to see the
> the relations not <identitly ...>. Now if I cut and paste
> 
> gap> [ r^2, s^2, s^-1*r*s*r ];
> [ <identity ...>, <identity ...>, <identity ...> ]
> 
> Is there an function that I can apply to rel to force the reduced
> multiplication without the cut and paste? 
> 
> I tried copies and tests like
> 
> gap> rel[1]=Identity(d4);
> false
> 
> but even  this does not do the reduction before the test.
> 
> Thanks for any comments
>     Ron
> 
> 
> 
> 
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