Exercise 20, § 21

Let G be a finite group, H a subgroup of $G,\sigma$ a finite-dimensional representation of H, and $\rho$ the representation of G induced by $\sigma$. Let C be a conjugacy class of G; the set $C\cap H$ is the disjoint union of a finite family $(D_i)_{i\in I}$ of conjugacy classes of H.

a) Prove the formula $\chi_{\rho}(C) =\frac{(G:H)}{Card(C)}\sum_{i\in I}$ Card(D$_i$)$\chi_{\sigma}(D_i)$.

b) If $\sigma$ is the trivial representation, then $\chi_{\rho}(C)=(G : H)$ Card(C)$^{-1}$Card(C $\cap H$) holds.