[GAP Forum] About regular and conjugacy actions

[saad]

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>