Question attached

need help with transitivety

If G acts on X and on Y, show that the formula g((x, y)) = (g(x), g(y)) defines an action of G on X x Y. Check that the stabilizer of (x, y) is the intersection of G, and G. Give an example which shows this action need not be transitive even if G acts transitively on both X and Y. We shall call this action the diagonal action of G on X x Y