(a) State Mackey's theorem, defining carefully all the terms used in the statement.

(b) Let $G$ be a finite group and suppose that $G$ acts on the set $\Omega$.

If $n \in \mathbb{N}$, we say that the action of $G$ on $\Omega$ is $n$-transitive if $\Omega$ has at least $n$ elements and for every pair of $n$-tuples $\left(a_{1}, \ldots, a_{n}\right)$ and $\left(b_{1}, \ldots, b_{n}\right)$ such that the $a_{i}$ are distinct elements of $\Omega$ and the $b_{i}$ are distinct elements of $\Omega$, there exists $g \in G$ with $g a_{i}=b_{i}$ for every $i$.

(i) Let $\Omega$ have at least $n$ elements, where $n \geqslant 1$ and let $\omega \in \Omega$. Show that $G$ acts $n$-transitively on $\Omega$ if and only if $G$ acts transitively on $\Omega$ and the stabiliser $G_{\omega}$ acts $(n-1)$-transitively on $\Omega \backslash\{\omega\}$.

(ii) Show that the permutation module $\mathbb{C} \Omega$ can be decomposed as

$$\mathbb{C} \Omega=\mathbb{C}_{G} \oplus V,$$

where $\mathbb{C}_{G}$ is the trivial module and $V$ is some $\mathbb{C} G$-module.

(iii) Assume that $|\Omega| \geqslant 2$, so that $V \neq 0$. Prove that $V$ is irreducible if and only if $G$ acts 2-transitively on $\Omega$. In that case show also that $V$ is not the trivial representation. [Hint: Pick any orbit of $G$ on $\Omega$; it is isomorphic as a $G$-set to $G / H$ for some subgroup $H \leqslant G$. Consider the induced character $\left.\operatorname{Ind}_{H}^{G} 1_{H} \cdot\right]$