Dear Vipul, For the specific action you describe (conjugation action on the set of subgroups), the short answer is to just count the number of conjugacy classes of subgroups. For example:
gap> g:=AlternatingGroup(4);; gap> cc:=ConjugacyClassesSubgroups(g);; gap> Size(cc); 5 (N.B. This count includes the "trivial" classes ()^G and G^G.) If you want, you can ask GAP to describe the structure of representatives of these conjugacy classes of subgroups: gap> List(cc, x->StructureDescription(Representative(x))); [ "1", "C2", "C3", "C2 x C2", "A4" ] I'm sure it's also possible to compute the number of orbits of subgroups under conjugation action by actually defining the action group and using the Orbits command. Maybe someone else will reply and describe that. As for the action of a group on the set of cosets of a subgroup, of course, this always results in a single orbit, so maybe isn't the action you're interested in... unless you consider this action on blocks of imprimitivity, in which case you do have orbits (the systems of imprimitivity). Here's an example: gap> G:=SymmetricGroup(4);; gap> cc:=ConjugacyClassesSubgroups(G);; gap> H:=Representative(cc[3]); # Group([ (3,4) ]) gap> act:=Action(G,RightCosets(G,H),OnRight); gap> blks:=AllBlocks(act); [ [ 1, 2, 3 ], [ 1, 4, 10 ], [ 1, 5, 9, 11 ], [ 1, 9 ] ] To see the full orbits (systems of imprimitivity), use gap> Orbit(act,blks[1],OnSets); [ [ 1, 2, 3 ], [ 7, 8, 9 ], [ 10, 11, 12 ], [ 4, 5, 6 ] ] gap> Orbit(act,blks[2],OnSets); [ [ 1, 4, 10 ], [ 2, 5, 8 ], [ 6, 9, 12 ], [ 3, 7, 11 ] ] etc. Hope that helps! -William On Sun, May 15, 2011 at 8:53 PM, vipul kakkar <vplkak...@gmail.com> wrote: > Dear all > Can I do following in GAP and how: > > Let G be a group and X be a set on which G acts, I want to calculate number > of orbits . > for example : Let G be any(finite) group and X the set of all subgroups of G > and G acts on X by conjugation and I want to compute number of orbit w.r.t > this action. > > Thanks > _______________________________________________ > Forum mailing list > Forum@mail.gap-system.org > http://mail.gap-system.org/mailman/listinfo/forum > _______________________________________________ Forum mailing list Forum@mail.gap-system.org http://mail.gap-system.org/mailman/listinfo/forum
