[GAP Forum] About regular and conjugacy actions

[Thomas Breuer]

Dear Saad Owaid,

the GAP commands which you show do not fit to your explanations.

- The call 'OnSets( AsSet( s ), s )' returns the list of conjugacy classes
  of the group 's'.
  I would say that this call is a misuse of the function 'OnSets',
  its second argument should be a group element not a group.

  The permutation action of the group 's' on its elements via conjugation
  can be obtained for example as 'Action( s, AsList( s ), OnPoints )'.

  (Your call 'Action( s, AsList( s ), OnRight )' does indeed compute
  the regular permutation action of the group 's'.)

- The calls 'OrbitStabilizer( s, gens[1], ract )' and
  'OrbitStabilizer( s, gens[1], cact )' do not make sense.
  The third argument is ignored in both cases,
  hence it is clear that the two results are equal.

  In order to compute orbit and stabilizer of a point 'x', say,
  under right multiplication and conjugation action, respectively,
  of the permutation group 's',
  one can call 'OrbitStabilizer( s, x, OnRight )' and
  'OrbitStabilizer( s, x, OnPoints )', respectively.

  (The results are different in your example.)

Perhaps the chapter "Groups and Homomorphisms" in the GAP Tutorial
helps to understand the ideas.

All the best,
Thomas

On Tue, Jun 09, 2020 at 07:33:26PM +0000, saad 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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