Finding right transversals using GAP

Question

I am new to the GAP software and I am following the GAP manual. I have found one right transversal to a subgroup of a group. Now i want to know if we can find all the right transversals of a subgroup in a group. If yes, how can we do so?

Answer 1

Different transversals just differ by changing representatives by different elements of the subgroup. It thus is sufficient to take one transversal, and multiply in all possible ways with elements of the subgroup. E.g.

gap> G:=Group((1,2,3),(1,4)(2,5));
Group([ (1,2,3), (1,4)(2,5) ])
gap> S:=Stabilizer(G,5);
Group([ (1,2,3), (1,4,2) ])
gap> t:=RightTransversal(G,S);
RightTransversal(Group([ (1,2,3), (1,4)(2,5) ]),Group([ (1,2,3),(1,4,2) ]))
gap> tups:=Tuples(AsSet(S),Length(t));;
gap> Length(tups);
248832
gap> alltrans:=List(tups,tup->List([1..Length(t)],x->tup[x]*t[x]));;

Note that storing all the elements in different transversals can quickly become quite memory intensive.