(i) Let $A_{4}$ denote the alternating group of even permutations of four symbols. Let $X$ be the 3-cycle $(123)$ and $P, Q$ be the pairs of transpositions $(12)(34)$ and $(13)(24)$. Find $X^{3}, P^{2}, Q^{2}, X^{-1} P X, X^{-1} Q X$, and show that $A_{4}$ is generated by $X, P$ and $Q$.

(ii) Let $G$ and $H$ be groups and let

$$G \times H=\{(g, h): g \in G, h \in H\}$$

Show how to make $G \times H$ into a group in such a way that $G \times H$ contains subgroups isomorphic to $G$ and $H$.

If $D_{n}$ is the dihedral group of order $n$ and $C_{2}$ is the cyclic group of order 2 , show that $D_{12}$ is isomorphic to $D_{6} \times C_{2}$. Is the group $D_{12}$ isomorphic to $A_{4}$ ?