Define W= <(1 2)(3 4)>, the cyclic subgroup of $\displaystyle S_4$ generated by (1 2)(3 4). Show that W is a normal subgroup of V, where V ={(1), (1 2)(3 4), (1 3)(2 4), (1 4)(2 3)}, but that W is not a normal subgroup of $\displaystyle S_4 $. Conclude that normality is not transitive: K is normal to H, and H is normal to G does not imply K is normal to G.