So far our approach to induction has been strictly based on characters. We proved using Frobenius reciprocity that if $\chi$ is the character of a representation $(\pi, V)$ of the subgroup $H$ of the finite group $G$, then $\chi^G$ is the character of a representation $(\pi^G, V^G)$ of the group $G$, where $\chi^G$ is given by the explicit formula

\[ \chi^G (g) = \frac{1}{|H|} \sum_{x \in G} \dot{\chi} (x g x^{- 1}) . \] | (4.04.0.1) |

Since we have only a little time left, we will be a little sketchy about details. Some of these will be put into Exercises. These have a different meaning from the Exercises in earlier chapters. Those were problems that I hoped you would write up and turn in. They were usually not integral parts of the text. In this Chapter, Exercises include facts that I don't fully have time to cover or write up. By doing these you will fill gaps in the exposition.