[GAP Forum] Problem with orbits on RightTransversal

[Anvita]

Ah, I see. I was misguided by the following sentence in 
Section 37.8 of the Manual:

>Functions that implement group actions such as Action or Permutation 
>(see Chapter Group Actions) use PositionCanonical, therefore it is possible to ``act'' 
>on a right transversal to implement the action on the cosets. This is often much 
>more efficient than acting on cosets.

So I thought that, in my case, Orbits would return the orbits with respect to that action
instead of generating an S3-set. The distinction can already be seen 
in the following code:

gap> 
gap> List(Orbits(Action(S3,RT,OnRight)),Size);
[ 3, 3 ]
gap> List(Orbits(S3,RT,OnRight),Size);        
[ 6, 6 ]
gap> 

A very subtle difference...

Anvita.

>
>While it looks strange, this is the intended and documented behavior.
>
>In your first example, your code is asking for the orbits of S3 on 
>the S3-set generated by RT.  This set has 12 elements and is not a 
>terribly interesting set.  The elements of RT are elements of S4, not 
>cosets and not equivalence classes of elements.
>
>In your second example your code is asking for the orbits of S3 on 
>the S3-set generated by RC.  This set is equal to RC and has 6 
>elements, each of which is a coset.
>
>There was a similar question on the forum earlier about actions on 
>conjugacy classes versus conjugacy class representatives, so perhaps 
>it would be good to put this sort of question on the FAQ.
>
>Anvita wrote:
>> Dear forum,
>>
>> The first part of the following code returns a strange result:
>> two orbits of size 6 on a 6-element right transversal.
>> Shouldn't the given action coincide with the action on
>> the right cosets as shown in the second part of the code?
>>
>> Thank you,
>> Anvita
>>
>> ----------------------------------------------------------------
>> gap>
>> gap> S4:=SymmetricGroup(4);
>> Sym( [ 1 .. 4 ] )
>> gap> S3:=SymmetricGroup(3);
>> Sym( [ 1 .. 3 ] )
>> gap> K:=Group((1,2),(1,2)(3,4));
>> Group([ (1,2), (1,2)(3,4) ])
>> gap> RT:=RightTransversal(S4,K);
>> RightTransversal(Sym( [ 1 .. 4 ] ),Group([ (1,2), (1,2)(3,4) ]))
>> gap> ORT:=Orbits(S3,RT,OnRight);;
>> gap> List(ORT,Size);
>> [ 6, 6 ]
>> gap>
>> gap>############################################################
>> gap>
>> gap> RC:=RightCosets(S4,K);;
>> gap> ORC:=Orbits(S3,RC,OnRight);;
>> gap> List(ORC,Size);
>> [ 3, 3 ]
>> gap>
>> ----------------------------------------------------------------
>>
>


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