In this question, all vector spaces will be complex.

(a) Let $A$ be a finite abelian group.

(i) Show directly from the definitions that any irreducible representation must be one-dimensional.

(ii) Show that $A$ has a faithful one-dimensional representation if and only if $A$ is cyclic.

(b) Now let $G$ be an arbitrary finite group and suppose that the centre of $G$ is nontrivial. Write $Z=\{z \in G \mid z g=g z \quad \forall g \in G\}$ for this centre.

(i) Let $W$ be an irreducible representation of $G$. Show that $\operatorname{Res}_{Z}^{G} W=\operatorname{dim} W \cdot \chi$, where $\chi$ is an irreducible representation of $Z$.

(ii) Show that every irreducible representation of $Z$ occurs in this way.

(iii) Suppose that $Z$ is not a cyclic group. Show that there does not exist an irreducible representation $W$ of $G$ such that every irreducible representation $V$ occurs as a summand of $W^{\otimes n}$ for some $n$.