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

[Hongyi Zhao]

On Sat, Apr 30, 2022 at 5:19 PM Dima Pasechnik <dima at sagemath.org> wrote:
>
> On Sat, Apr 30, 2022 at 05:04:57PM +0800, Hongyi Zhao wrote:
> > 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.
>
> rather, an extra ')' - sorry for typo. It's not needed:
>
> gap> f := FreeGroup( "a");
> <free group on the generators [ a ]>
> gap> g:=f/[f.1^7];
> <fp group on the generators [ a ]>

Nice. Thank you for pointing out this tip.

> Best,
> Dima

Yours,
Hongyi