(a) For any finite group $G$, let $\rho_{1}, \ldots, \rho_{k}$ be a complete set of non-isomorphic complex irreducible representations of $G$, with dimensions $n_{1}, \ldots n_{k}$, respectively. Show that

$$\sum_{j=1}^{k} n_{j}^{2}=|G|$$

(b) Let $A, B, C, D$ be the matrices

and let $G=\langle A, B, C, D\rangle$. Write $Z=-I_{4}$.

(i) Prove that the derived subgroup $G^{\prime}=\langle Z\rangle$.

(ii) Show that for all $g \in G, g^{2} \in\langle Z\rangle$, and deduce that $G$ is a 2-group of order at most 32 .

(iii) Prove that the given representation of $G$ of degree 4 is irreducible.

(iv) Prove that $G$ has order 32 , and find all the irreducible representations of $G$.

$$\begin{aligned} & A=\left(\begin{array}{cccc}1 & 0 & 0 & 0 \\0 & -1 & 0 & 0 \\0 & 0 & 1 & 0 \\0 & 0 & 0 & -1\end{array}\right), \quad B=\left(\begin{array}{llll}0 & 1 & 0 & 0 \\1 & 0 & 0 & 0 \\0 & 0 & 0 & 1 \\0 & 0 & 1 & 0\end{array}\right) \\ & C=\left(\begin{array}{cccc}1 & 0 & 0 & 0 \\0 & 1 & 0 & 0 \\0 & 0 & -1 & 0 \\0 & 0 & 0 & -1\end{array}\right), \quad D=\left(\begin{array}{llll}0 & 0 & 1 & 0 \\0 & 0 & 0 & 1 \\1 & 0 & 0 & 0 \\0 & 1 & 0 & 0\end{array}\right) \end{aligned}$$