[GAP Forum] Construct the C8 from finitely presented group and check its isomorphism with permutation group.

[Hongyi Zhao]

On Sat, Apr 30, 2022 at 4:49 PM Dima Pasechnik <dima at sagemath.org> wrote:
>
> On Sat, Apr 30, 2022 at 11:27:37AM +0800, Hongyi Zhao wrote:
> > Hi GAP team,
> >
> > I try to construct the C8 from finitely presented group and check its
> > isomorphism with permutation group with the following steps:
> >
> > gap> f := FreeGroup( "a");
> > <free group on the generators [ a ]>
> >
> > gap> g:=f/[ f.1, f.1^2, f.1^3, f.1^4, f.1^5, f.1^6, f.1^7 ];
> > <fp group on the generators [ a ]>
> >
> > gap> h:=IsomorphismPermGroup(g);
> > [ a ] -> [ () ]
> >
> > Is there any problem with my operations?
>
> It's correct, as your g is a trivial group (as you take the quotient over the whole group f)
>
> To get a finitely presented C8, do
>
> g:=f/[f.1^7]);

Thank you for pointing this out. I want to add some additional remarks:

1. There is a missing `(` in your above code.
2. To get a finitely presented C8, the following should be used:

g:=f/[f.1^8]);

Please see my related tests below:

gap> f := FreeGroup( "a");
gap> g:=f/([f.1^7]);
<fp group on the generators [ a ]>
gap> Elements(g);
[ <identity ...>, a, a^6, a^2, a^5, a^3, a^4 ]
gap> StructureDescription(g);
"C7"
gap> g:=f/([f.1^8]);
<fp group on the generators [ a ]>
gap> StructureDescription(g);
"C8"
gap> Elements(g);
[ <identity ...>, a, a^2, a^4, a^3, a^-3, a^-2, a^-1 ]

> HTH
> Dima

Yours,
Hongyi