[GAP Forum] About regular and conjugacy actions

[saad]

Dear forum;
I found the following definition, Two permutation groups G, H ≤ Sym(Ω) are called orbit equivalent if they havethe same orbits on the power set of Ω. Primitive orbit equivalent permutation groupswere determined by Seress. Is this explains my computations!
What if such a case happens on actions different than a power set of Ω can one generalize the definition to be orbit equivalent if they have the same orbits on any domain of action.  
Thanks,
Saad Owaid.    On Tuesday, June 9, 2020, 10:34:12 PM GMT+3, saad <saadhala10 at hotmail.com> wrote:  
 
 Dear all,
I made a simple calculations to compare orbits and stabilizers of symmetric group via its regular and conjugacy actions
I find that they are always the same, can anyone help to explain, is it always this situation ..any comment (s) are (is) realy appreciated.
thanks.
Saad Owaid
Here what I did:
gap> s:=SymmetricGroup(5);;gap> gens:=GeneratorsOfGroup(s);[ (1,2,3,4,5), (1,2) ]gap>  ract:=Action(s,AsList(s),OnRight);<permutation group with 2 generators>gap> cact:=OnSets(AsSet(s),s);[ ()^G, (4,5)^G, (2,3,5)^G, (2,3)(4,5)^G, (1,3,5,2)^G, (1,4,2)(3,5)^G, (1,5,4,3,2)^G ]gap> orbstab:=OrbitStabilizer(s,gens[1],ract);rec( orbit := [ (1,2,3,4,5), (1,3,4,5,2), (1,3,2,4,5), (1,2,4,3,5), (1,4,5,2,3), (1,2,3,5,4), (1,4,3,5,2), (1,3,4,2,5),      (1,5,2,3,4), (1,3,5,4,2), (1,3,2,5,4), (1,2,4,5,3), (1,5,3,2,4), (1,5,2,4,3), (1,5,4,2,3), (1,4,2,3,5), (1,4,5,3,2),      (1,4,2,5,3), (1,4,3,2,5), (1,5,3,4,2), (1,2,5,3,4), (1,3,5,2,4), (1,2,5,4,3), (1,5,4,3,2) ], stabilizer := Group([ (1,2,3,4,   5) ]) )gap> orbstab:=OrbitStabilizer(s,gens[1],cact);rec( orbit := [ (1,2,3,4,5), (1,3,4,5,2), (1,3,2,4,5), (1,2,4,3,5), (1,4,5,2,3), (1,2,3,5,4), (1,4,3,5,2), (1,3,4,2,5),      (1,5,2,3,4), (1,3,5,4,2), (1,3,2,5,4), (1,2,4,5,3), (1,5,3,2,4), (1,5,2,4,3), (1,5,4,2,3), (1,4,2,3,5), (1,4,5,3,2),      (1,4,2,5,3), (1,4,3,2,5), (1,5,3,4,2), (1,2,5,3,4), (1,3,5,2,4), (1,2,5,4,3), (1,5,4,3,2) ], stabilizer := Group([ (1,2,3,4,   5) ]) )gap> orbstab:=OrbitStabilizer(s,gens[2],ract);rec( orbit := [ (1,2), (2,3), (3,4), (1,3), (4,5), (2,4), (1,5), (3,5), (1,4), (2,5) ], stabilizer := Group([ (1,2), (4,5), (3,   4) ]) )gap> orbstab:=OrbitStabilizer(s,gens[2],cact);rec( orbit := [ (1,2), (2,3), (3,4), (1,3), (4,5), (2,4), (1,5), (3,5), (1,4), (2,5) ], stabilizer := Group([ (1,2), (4,5), (3,   4) ]) )gap>

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