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: > > > hul...@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: > hul...@colostate.edu, http://www.math.colostate.edu/~hulpke > >
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