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

[Hongyi Zhao]

On Mon, May 2, 2022 at 2:36 AM Thomas Breuer <sam at math.rwth-aachen.de> wrote:
>
> 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 ) ]'.

Thank you for your explanation. I confirmed this 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_1:= f2 / [ f2.1^2, f2.2^2, Comm( f2.1, f2.2 ) ];
g4_2_2:= f2 / [ f2.1^2, f2.2^2, Comm( f2.1, f2.2 ) ];
Elements(G4_2); Elements(g4_2_1); Elements(g4_2_2);

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_1:= f2 / [ f2.1^2, f2.2^2, Comm( f2.1, f2.2 ) ];
<fp group on the generators [ P, Q ]>
gap> g4_2_2:= f2 / [ f2.1^2, f2.2^2, Comm( f2.1, f2.2 ) ];
<fp group on the generators [ P, Q ]>
gap> Elements(G4_2); Elements(g4_2_1); Elements(g4_2_2);
[ <identity ...>, P, Q, P*Q ]
[ <identity ...>, P, Q, P*Q ]
[ <identity ...>, P, Q, P*Q ]
gap>


Side additional remarks:

1. I try to query the built-in help with the following command, but
obtained so many entries I don't know how to choose from:

gap> ?Comm
   Choose an entry to view, 'q' for none (or use ?<nr> later):

         │
│[1]   Guarana (not loaded): Comm

          │
│[2]   Utils: Comm

          │
│[3]   ANUPQ (not loaded): Commands from the Main p-Quotient menu
                                                                [...]


2. I also tried with the same method to check the following two commands:

gap> ?quit
gap> ?QUIT

Gap gives exactly the same document for them:

> `quit'{quit}
    This  closes  all  windows and just brings you back to the GAP
window. In a new start of ITC all switches will be set to their
default values.
    The group that you had handled will still be available as a GAP object.


But I confirmed the following differences between them based on the
description in the GAP reference manual:

1. quit

type 'quit;' to quit to outer loop

2. In manual, the following description is used:

6.7.1
QUIT
. QUIT
(global variable)
An emergency way to leave GAP is to enter QUIT at any gap> or brk> or
brk_nn > prompt.


> All the best
> Thomas

Yours,
Hongyi

> 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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