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

[Hongyi Zhao]

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