(a) Let $G$ be a finite group and $X$ a finite set on which $G$ acts. Define the permutation representation $\mathbb{C}[X]$ and compute its character.

(b) Let $G$ and $U$ be the following subgroups of $\mathrm{GL}_{2}\left(\mathbb{F}_{p}\right)$, where $p$ is a prime,

$$G=\left\{\left(\begin{array}{cc} a & b \\ 0 & 1 \end{array}\right) \mid a \in \mathbb{F}_{p}^{\times}, b \in \mathbb{F}_{p}\right\}, \quad U=\left\{\left(\begin{array}{ll} 1 & b \\ 0 & 1 \end{array}\right) \mid b \in \mathbb{F}_{p}\right\}$$

(i) Decompose $\mathbb{C}[G / U]$ into irreducible representations.

(ii) Let $\psi: U \rightarrow \mathbb{C}^{\times}$be a non-trivial, one-dimensional representation. Determine the character of the induced representation $\operatorname{Ind}_{U}^{G} \psi$, and decompose $\operatorname{Ind}_{U}^{G} \psi$ into irreducible representations.

(iii) List all of the irreducible representations of $G$ and show that your list is complete.