Proposition 3

Let H be a subgroup of G of finite index. Let $(M, \sigma )$ be a linear representation of H. Suppose that M is a finite-dimensional free K-module. Denote by $(V, \pi )$ the representation of G induced by $\sigma$. Then the K-module V is free and finite-dimensional, and

(5) $\chi_{\pi}=$ Ind$^G_H(\chi_{\sigma})$.

Let S be a system of representatives of $G/H$. As we saw above, the linear mapping $M^S\rightarrow$ Ind$^G_H(M)$ given by $(m_s)_{s\in S}\mapsto \sum_{s\in S}s\otimes m_s$ is a K-module isomorphism. In particular, V is a finite-dimensional free K-module. For any $g\in G$, denote by $M_g$ the image of M by the mapping $m\mapsto g\otimes m$. For all $g, g'\in G$, we have $M_g= M_{g'}$ if and only if $gH =g'H$, and V is the direct sum of the submodules $M_s$ for $s\in S$. For all $g, g'\in G$, we also have $\pi (g)M_{g'}= M_{gg'}$. For any $g\in G$, denote by $S_g$ the set of elements $s$ of S such that $s^{-1}gs\in H$. For all $g, g'\in G$ such that $g^{'-1}gg'\in H$, the automorphism $\pi (g)$ of V induces an automorphism $\pi (g)_{g'}$ of $M_{g'}$, and we have

Tr($\pi (g)$) $=\sum_{s\in S_g}$ Tr($\pi (g)_s$) $=\sum_{s\in S_g}$ Tr($\sigma (s^{-1}gs)$).

The last assertion of the proposition follows.