Exercise 4, § 21

Let $(M, \pi )$ be a faithful[^1] finite-dimensional representation of G and $(N, \rho )$ an irreducible representation; suppose that the character $\chi_{\pi}$ has exactly $s$ distinct values. Prove that there exists an integer $n < s$ such that the K[G]-module N is isomorphic to a submodule of $M^{\otimes n}$ (calculate $\langle \chi_{\rho}, \chi^n_{\pi}\rangle$, observing that the equality $\chi_{\pi}(g) =$ dim M is equivalent to $g= 1$).

[^1]: This means that the homomorphism $\pi : G\rightarrow$ Aut$_K(M)$ is injective.