Let $G$ be a finite group and $Z$ its centre. Suppose that $G$ has order $n$ and $Z$ has order $m$. Suppose that $\rho: G \rightarrow \mathrm{GL}(V)$ is a complex irreducible representation of degree $d .$

(i) For $g \in Z$, show that $\rho(g)$ is a scalar multiple of the identity.

(ii) Deduce that $d^{2} \leqslant n / m$.

(iii) Show that, if $\rho$ is faithful, then $Z$ is cyclic.

[Standard results may be quoted without proof, provided they are stated clearly.]

Now let $G$ be a group of order 18 containing an elementary abelian subgroup $P$ of order 9 and an element $t$ of order 2 with $t x t^{-1}=x^{-1}$ for each $x \in P$. By considering the action of $P$ on an irreducible $\mathbb{C} G$-module prove that $G$ has no faithful irreducible complex representation.