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

[Hongyi Zhao]

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 ]

2. Why does the IsomorphismPermGroup(g) give the following result?

[ (1,2,4,6,8,7,5,3) ]


[1] https://www.gap-system.org/Manuals/doc/ref/chap47.html#X7AA982637E90B35A

Regards
Hongyi