In this question all representations are complex and $G$ is a finite group.

(a) State and prove Mackey's theorem. State the Frobenius reciprocity theorem.

(b) Let $X$ be a finite $G$-set and let $\mathbb{C} X$ be the corresponding permutation representation. Pick any orbit of $G$ on $X$ : it is isomorphic as a $G$-set to $G / H$ for some subgroup $H$ of $G$. Write down the character of $\mathbb{C}(G / H)$.

(i) Let $\mathbb{C}_{G}$ be the trivial representation of $G$. Show that $\mathbb{C} X$ may be written as a direct sum

$$\mathbb{C} X=\mathbb{C}_{G} \oplus V$$

for some representation $V$.

(ii) Using the results of (a) compute the character inner product $\left\langle 1_{H} \uparrow^{G}, 1_{H} \uparrow^{G}\right\rangle_{G}$ in terms of the number of $(H, H)$ double cosets.

(iii) Now suppose that $|X| \geqslant 2$, so that $V \neq 0$. By writing $\mathbb{C}(G / H)$ as a direct sum of irreducible representations, deduce from (ii) that the representation $V$ is irreducible if and only if $G$ acts 2 -transitively. In that case, show that $V$ is not the trivial representation.