Exercise 9, § 21

Let $(M, \pi )$ be an irreducible representation of G and H a normal subgroup of G. a) Let $M =\bigoplus_{i\in I}M_i$ be the decomposition of the K[H]-module M as a direct sum of isotypical submodules. Prove that the action of G on M permutes the submodules $M_i$ transitively. Deduce that $(M, \pi )$ is isomorphic to the representation induced by a subgroup of G containing H (apply Exercise 5, b)).

b) Let $\mathscr{S}$ be the support of the K[H]-module M; there exists an integer $e$ such that M is K[H]-isomorphic to $\sum_{\lambda\in\mathscr{S}}\lambda^e$.