(i) State Frobenius' theorem for transitive permutation groups acting on a finite set. Define Frobenius group and show that any finite Frobenius group (with an appropriate action) satisfies the hypotheses of Frobenius' theorem.

(ii) Consider the group

$$F_{p, q}:=\left\langle a, b: a^{p}=b^{q}=1, b^{-1} a b=a^{u}\right\rangle$$

where $p$ is prime, $q$ divides $p-1$ ( $q$ not necessarily prime), and $u$ has multiplicative order $q$ modulo $p$ (such elements $u$ exist since $q$ divides $p-1)$. Let $S$ be the subgroup of $\mathbb{Z}_{p}^{\times}$ consisting of the powers of $u$, so that $|S|=q$. Write $r=(p-1) / q$, and let $v_{1}, \ldots, v_{r}$ be coset representatives for $S$ in $\mathbb{Z}_{p}^{\times}$.

(a) Show that $F_{p, q}$ has $q+r$ conjugacy classes and that a complete list of the classes comprises $\{1\},\left\{a^{v_{j} s}: s \in S\right\}(1 \leqslant j \leqslant r)$ and $\left\{a^{m} b^{n}: 0 \leqslant m \leqslant p-1\right\}(1 \leqslant n \leqslant q-1)$.

(b) By observing that the derived subgroup $F_{p, q}^{\prime}=\langle a\rangle$, find $q$ 1-dimensional characters of $F_{p, q}$. [Appropriate results may be quoted without proof.]

(c) Let $\varepsilon=e^{2 \pi i / p}$. For $v \in \mathbb{Z}_{p}^{\times}$denote by $\psi_{v}$ the character of $\langle a\rangle$ defined by $\psi_{v}\left(a^{x}\right)=\varepsilon^{v x}(0 \leqslant x \leqslant p-1)$. By inducing these characters to $F_{p, q}$, or otherwise, find $r$ distinct irreducible characters of degree $q$.