(a) Suppose $H$ is a subgroup of a finite group $G, \chi$ is an irreducible character of $G$ and $\varphi_{1}, \ldots, \varphi_{r}$ are the irreducible characters of $H$. Show that in the restriction $\chi \downarrow_{H}=a_{1} \varphi_{1}+\cdots+a_{r} \varphi_{r}$, the multiplicities $a_{1}, \ldots, a_{r}$ satisfy

$$\sum_{i=1}^{r} a_{i}^{2} \leqslant|G: H|$$

Determine necessary and sufficient conditions under which the inequality in ( $\uparrow$ ) is actually an equality.

(b) Henceforth suppose that $H$ is a (normal) subgroup of index 2 in $G$, and that $\chi$ is an irreducible character of $G$.

Lift the non-trivial linear character of $G / H$ to obtain a linear character of $G$ which satisfies

$$\lambda(g)= \begin{cases}1 & \text { if } g \in H \\ -1 & \text { if } g \notin H\end{cases}$$

(i) Show that the following are equivalent:

(1) $\chi \downarrow_{H}$ is irreducible;

(2) $\chi(g) \neq 0$ for some $g \in G$ with $g \notin H$;

(3) the characters $\chi$ and $\chi \lambda$ of $G$ are not equal.

(ii) Suppose now that $\chi \downarrow_{H}$ is irreducible. Show that if $\psi$ is an irreducible character of $G$ which satisfies

$$\psi \downarrow_{H}=\chi \downarrow_{H}$$

then either $\psi=\chi$ or $\psi=\chi \lambda .$

(iii) Suppose that $\chi \downarrow_{H}$ is the sum of two irreducible characters of $H$, say $\chi \downarrow_{H}=\psi_{1}+\psi_{2}$. If $\phi$ is an irreducible character of $G$ such that $\phi \downarrow_{H}$ has $\psi_{1}$ or $\psi_{2}$ as a constituent, show that $\phi=\chi$.

(c) Suppose that $G$ is a finite group with a subgroup $K$ of index 3 , and let $\chi$ be an irreducible character of $G$. Prove that

$$\left\langle\chi \downarrow_{K}, \chi \downarrow_{K}\right\rangle_{K}=1,2 \text { or } 3$$

Give examples to show that each possibility can occur, giving brief justification in each case.