Define the concepts of induction and restriction of characters. State and prove the Frobenius Reciprocity Theorem.

Let $H$ be a subgroup of $G$ and let $g \in G$. We write $\mathcal{C}(g)$ for the conjugacy class of $g$ in $G$, and write $C_{G}(g)$ for the centraliser of $g$ in $G$. Suppose that $H \cap \mathcal{C}(g)$ breaks up into $m$ conjugacy classes of $H$, with representatives $x_{1}, x_{2}, \ldots, x_{m}$.

Let $\psi$ be a character of $H$. Writing $\operatorname{Ind}_{H}^{G}(\psi)$ for the induced character, prove that

(i) if no element of $\mathcal{C}(g)$ lies in $H$, then $\operatorname{Ind}_{H}^{G}(\psi)(g)=0$,

(ii) if some element of $\mathcal{C}(g)$ lies in $H$, then

$$\operatorname{Ind}_{H}^{G}(\psi)(g)=\left|C_{G}(g)\right| \sum_{i=1}^{m} \frac{\psi\left(x_{i}\right)}{\left|C_{H}\left(x_{i}\right)\right|} .$$

Let $G=S_{4}$ and let $H=\langle a, b\rangle$, where $a=\left(\begin{array}{llll}1 & 2 & 3 & 4\end{array}\right)$ and $b=\left(\begin{array}{l}1\end{array}\right.$ dihedral group and write down its character table. Restrict each $G$-conjugacy class to $H$ and calculate the $H$-conjugacy classes contained in each restriction. Given a character $\psi$ of $H$, express Ind ${ }_{H}^{G}(\psi)(g)$ in terms of $\psi$, where $g$ runs through a set of conjugacy classes of $G$. Use your calculation to find the values of all the irreducible characters of $H$ induced to $G$.

$$\begin{aligned} & a=\left(\begin{array}{lllll}1 & 2 & 3 & 5 & 6\end{array}\right)(789101112), \\ & b=\left(\begin{array}{llll}1 & 7 & 4 & 10\end{array}\right)\left(\begin{array}{llll}2 & 12 & 5 & 9\end{array}\right)(368), \end{aligned}$$