[GAP Forum] FactorGroup of permutation groups

[Hulpke,Alexander]

> 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