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

[Hongyi Zhao]

On Sat, Apr 30, 2022 at 4:54 PM Dima Pasechnik <dima at sagemath.org> wrote:
>
> On Sat, Apr 30, 2022 at 01:20:01PM +0800, Hongyi Zhao wrote:
> > On Sat, Apr 30, 2022 at 11:27 AM Hongyi Zhao <hongyi.zhao at gmail.com> 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?
> >
> > Based on the example given in the GAP - Reference Manual [1], I
> > figured out the following steps:
> >
> > gap> f := FreeGroup( "a");
> > <free group on the generators [ a ]>
> > gap> g:=f/[ [ f.1^-1,  f.1^7] ];
> > <fp group on the generators [ a ]>
> > gap> Elements( g );
> > [ <identity ...>, a, a^7, a^2, a^6, a^3, a^5, a^4 ]
> > gap> StructureDescription( g );
> > "C8"
> > gap> IsomorphismPermGroup(g);
> > [ a ] -> [ (1,2,4,6,8,7,5,3) ]
> >
> >
> > But I still have the following puzzles:
> >
> > 1. Why are the group elements not displayed in the following order?
> >
> > [ <identity ...>, a, a^2, a^3, a^4, a^5, a^6, a^7 ]
>
> there is no guarantee for group elements to be shown in any particular order.

Thank you for your explanation. Neither the order of group elements,
nor the actual form of them, are fixed. See my following example for
more information:

gap> f := FreeGroup( "a");
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 ]


> >
> > 2. Why does the IsomorphismPermGroup(g) give the following result?
> >
> > [ (1,2,4,6,8,7,5,3) ]
>
> This means that replacing a with this permutation gives you an insomorphic permutation group.

Could you please explain this in more detail? Here we only have one
element, i.e., a, so, how to replace it?

> Do you mean to ask why it's not [(1,...,7)] ?

Exactly.

> There is no guarantee that it will pick up one's favourite generators.

Thank you for your in-depth explanation again.

> HTH
> Dima