Group action on direct product in GAP

Question

I want to define the group action of $G$ on $G \times G$ by left multiplication. For example I have G:=DihedralGroup(6);. I created L:=Elements(G); and M:=Cartesian(L,L);. I am not sure how to write function $G \times M \to M$ as $ g\cdot(h,k) \to (gh,gk)$ in GAP which is needed for OrbitsDomain function. How can I do it?

Answer 1

In GAP actions always have to be right actions, so you would have to have $(h,k)\cdot g\mapsto(hg,kg)$ so the action is consistent. (Or you could map to $(g^{-1}h,g^{-1}k)$.)

With this, you could define the action function as this:

OnPairsRight:=function(pair,g)
  return [pair[1]*g,pair[2]*g];
end;

and then use o:=OrbitsDomain(G,M,OnPairsRight);