(i) Let $G$ be a finite group. Show that

(1) If $\chi$ is an irreducible character of $G$ then so is its conjugate $\bar{\chi}$.

(2) The product of any two characters of $G$ is again a character of $G$.

(3) If $\chi$ and $\psi$ are irreducible characters of $G$ then

$$\left\langle\chi \psi, 1_{G}\right\rangle= \begin{cases}1, & \text { if } \chi=\bar{\psi}, \\ 0, & \text { if } \chi \neq \bar{\psi} .\end{cases}$$

(ii) If $\chi$ is a character of the finite group $G$, define $\chi_{S}$ and $\chi_{A}$. For $g \in G$ prove that

$$\chi_{S}(g)=\frac{1}{2}\left(\chi^{2}(g)+\chi\left(g^{2}\right)\right) \quad \text { and } \chi_{A}(g)=\frac{1}{2}\left(\chi^{2}(g)-\chi\left(g^{2}\right)\right)$$

(iii) A certain group of order 24 has precisely seven conjugacy classes with representatives $g_{1}, \ldots, g_{7}$; further, $G$ has a character $\chi$ with values as follows:

$$\begin{array}{cccccccc} g_{i} & g_{1} & g_{2} & g_{3} & g_{4} & g_{5} & g_{6} & g_{7} \\ \left|C_{G}\left(g_{i}\right)\right| & 24 & 24 & 4 & 6 & 6 & 6 & 6 \\ \chi & 2 & -2 & 0 & -\omega^{2} & -\omega & \omega & \omega^{2} \end{array}$$

where $\omega=e^{2 \pi i / 3}$.

It is given that $g_{1}^{2}, g_{2}^{2}, g_{3}^{2}, g_{4}^{2}, g_{5}^{2}, g_{6}^{2}, g_{7}^{2}$ are conjugate to $g_{1}, g_{1}, g_{2}, g_{5}, g_{4}, g_{4}, g_{5}$ respectively.

Determine $\chi_{S}$ and $\chi_{A}$, and show that both are irreducible.