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> >> ---------------------------------------------------------------- >> > = _______________________________________________ Forum mailing list Forum@mail.gap-system.org http://mail.gap-system.org/mailman/listinfo/forum
