(i) State the orbit-stabilizer theorem for a group $G$ acting on a set $X$.

(ii) Denote the group of all symmetries of the cube by $G$. Using the orbit-stabilizer theorem, show that $G$ has 48 elements.

Does $G$ have any non-trivial normal subgroups?

Let $L$ denote the line between two diagonally opposite vertices of the cube, and let

$$H=\{g \in G \mid g L=L\}$$

be the subgroup of symmetries that preserve the line. Show that $H$ is isomorphic to the direct product $S_{3} \times C_{2}$, where $S_{3}$ is the symmetric group on 3 letters and $C_{2}$ is the cyclic group of order 2 .