Exercise 5, § 21

Let $(M, \pi )$ be a representation of G. a) Let H be a subgroup of G, S a system of representatives in G of the left cosets (mod H), and N a linear subspace of M stable under H. The representation of G in M is isomorphic to the representation induced by that of H in N if and only if we have $M =\bigoplus_{s\in S}sN$. b) Suppose that M is the direct sum of a family $(M_i)_{i\in I}$ of subspaces and that there exists a transitive action of G on I such that $\pi (g)(M_i)\subset M_{gi}$ for $g\in G,i\in I$. Let $o\in I$, and let $G_o$ be its stabilizer in G. Prove that $\pi$ is isomorphic to the representation induced by the representation of $G_o$ in $M_o$. If $\pi$ is irreducible, then the condition that the action of G on I is transitive is automatically satisfied.