[GAP Forum] FactorGroup of permutation groups

[tkohl at math.bu.edu]

Fantastic, thank you .

On Tue, 8 Dec 2020, Hulpke,Alexander wrote:

>  
> > Dear Forum, Alexander,
> >
> > > Dear Forum, Dear Tim Kohl,
> > >
> > > If N, H are permutation groups with H normal in N
> > > and one computes FactorGroup(N,H) the result is expressed
> > > in terms of generators and relations.
> > >
> > > I suspect it is a PcGroup (which happens i factor is solvable). Otherwise it will be a permutation group.
> > >
> > > Is there a way to
> > > correlate the generators of FactorGroup(N,H) with a
> > > transversal of H in N?
> > >
> > > So you probably want the permutation action of N on the cosets of H. You can get it as `FactorCosetAction(N,H)` with the numbering of points corresponding to `RightTransversal(N,H)`.
> >
> > I guess my natural question (betraying a bit of ignorance) is how can I utilize this Action?
> >
> > And if I have a subgroup of FactorGroup(N,H) can I look at the resulting action on the level of cosets?
> 
> Maybe an example will be easiest. Let's take as Group SL_2(5) and its action on 2- Sylow subgroups:
> 
> gap> G:=SL(2,5);;
> gap> S:=SylowSubgroup(G,2);
> gap> H:=Normalizer(G,S);;
> gap> Index(G,H);
> 5
> 
> gap> T:=RightTransversal(G,H);
> RightTransversal(SL(2,5),Group([ [ [ Z(5), 0*Z(5) ], [ Z(5)^0, Z(5)^3 ] ],
>   [ [ Z(5), Z(5) ], [ 0*Z(5), Z(5)^3 ] ],
>   [ [ Z(5)^2, 0*Z(5) ], [ 0*Z(5), Z(5)^2 ] ],
>   [ [ Z(5)^0, Z(5)^2 ], [ Z(5)^0, 0*Z(5) ] ] ]))
> gap> act:=FactorCosetAction(G,H);;
> gap> f:=Image(act);;Size(f);
> 60
> 
> Now lets look at the correspondence with a random element:
> gap> elm:=Random(G);; # some element
> gap> Image(act,elm);
> (1,4,5)
> 
> So this element will map coset 4 to coset 5:  (and coset 1 to 4, and fix cosets 2 and 3).
> 
> gap> T[4]*elm/T[5] in H;
> true
> 
> Best,
>  
>   Alexander
> 
> >
> >
> > Thank you.
> >
> > -T
> >
> > >
> > > All the best,
> > >
> > > Alexander Hulpke
> > >
> > > The reason I'm asking is that N acts transitively
> > > on a collection of groups with H as the stabilizer
> > > and I want to study the induced action of N/H (and most
> > > importantly subgroups thereof) as it is substantially smaller.
> > >
> > > Thanks.
> > >
> > > -Tim K.
> > >
> > > - Colorado State University, Department of Mathematics, Weber Building, 1874 Campus Delivery, Fort Collins, CO 80523-1874, USA email: hulpke at colostate.edu, http://www.math.colostate.edu/~hulpke
> > >
> 
> - Colorado State University, Department of Mathematics, Weber Building, 1874 Campus Delivery, Fort Collins, CO 80523-1874, USA email: hulpke at colostate.edu, http://www.math.colostate.edu/~hulpke
> 
>