(a) What does it mean to say that a representation of a group is completely reducible? State Maschke's theorem for representations of finite groups over fields of characteristic 0 . State and prove Schur's lemma. Deduce that if there exists a faithful irreducible complex representation of $G$, then $Z(G)$ is cyclic.

(b) If $G$ is any finite group, show that the regular representation $\mathbb{C} G$ is faithful. Show further that for every finite simple group $G$, there exists a faithful irreducible complex representation of $G$.

(c) Which of the following groups have a faithful irreducible representation? Give brief justification of your answers.

(i) the cyclic groups $C_{n}(n$ a positive integer $)$;

(ii) the dihedral group $D_{8}$;

(iii) the direct product $C_{2} \times D_{8}$.