[GAP Forum] The equivalence of the two methods to create finitely presented groups based on the quotient of a free group.

[Thomas Breuer]

Dear Hongyi,

the entry '[ f2.1 , f2.1^-1 ]' means the equation 'f2.1 = f2.1^-1',
which is equivalent to the relator 'f2.1^2'.
Thus an equivalent description of your group 'G4_2' in terms of
defining relators would be 'f2 / [ f2.1^2, f2.2^2, Comm( f2.1, f2.2 ) ]'.

All the best
Thomas


On Sun, May 01, 2022 at 10:08:59PM +0800, Hongyi Zhao wrote:
> Hi GAP team,
> 
> In the chapter 47 of GAP - Reference Manual, the following description is used:
> 
> So to create a finitely presented group you first have to generate a
> free group (see FreeGroup (37.2.1) for details). There are two ways to
> specify a quotient of the free group: either by giving a list of
> relators or by giving a list of equations.
> 
> So, I try to verify the equivalence of the above two methods with the
> following code snippet:
> 
> f2 := FreeGroup("P", "Q");;
> G4_2:= f2/[ [ f2.1 , f2.1^-1 ], [ f2.2 , f2.2^-1 ], [ f2.2 *f2.1, f2.1
> *f2.2 ] ];
> g4_2:= f2/[ f2.1,f2.2,f2.1*f2.2];
> 
> Elements(G4_2);
> Elements(g4_2);
> 
> But I obtained the following results:
> 
> gap> f2 := FreeGroup("P", "Q");;
> gap> G4_2:= f2/[ [ f2.1 , f2.1^-1 ], [ f2.2 , f2.2^-1 ], [ f2.2 *f2.1,
> f2.1 *f2.2 ] ];
> <fp group on the generators [ P, Q ]>
> gap> g4_2:= f2/[ f2.1,f2.2,f2.1*f2.2];
> <fp group on the generators [ P, Q ]>
> gap>
> gap> Elements(G4_2);
> [ <identity ...>, P, Q, P*Q ]
> gap> Elements(g4_2);
> [ <identity ...> ]
> gap>
> 
> 
> As you can see, the groups obtained by the two methods are not
> equivalent. So, I want to know, can the groups generated by these two
> methods be isomorphic to each other?
> 
> Regards
> Hongyi (Hongsheng)
> -- 
> Assoc. Prof. Hongsheng Zhao <hongyi.zhao at gmail.com>
> Theory and Simulation of Materials
> Hebei Vocational University of Technology and Engineering
> No. 473, Quannan West Street, Xindu District, Xingtai, Hebei province
> 
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