(i) Define the character $\chi$ of a representation $D$ of a finite group $G$. Show that $<\chi \mid \chi>=1$ if and only if $D$ is irreducible, where

$$<\chi \mid \chi>=\frac{1}{|G|} \sum_{g \in G} \chi(g) \chi\left(g^{-1}\right)$$

If $|G|=8$ and $<\chi \mid \chi>=2$, what are the possible dimensions of the representation $D ?$

(ii) State and prove Schur's first and second lemmas.