(i) State the orbit-stabilizer theorem.

Let $G$ be the group of rotations of the cube, $X$ the set of faces. Identify the stabilizer of a face, and hence compute the order of $G$.

Describe the orbits of $G$ on the set $X \times X$ of pairs of faces.

(ii) Define what it means for a subgroup $N$ of $G$ to be normal. Show that $G$ has a normal subgroup of order 4 .