Exercise 15, § 21

Suppose that all representations of G are monomial; let us prove that G is solvable. Reasoning by induction on the order of G, we may assume that every quotient group of G different from G is solvable.

a) If G contains two distinct minimal nontrivial normal subgroups $H_1$ and $H_2$, then it is solvable (consider the canonical homomorphism from G to $G/H_1\times G/H_2$).

Assume from now on that G admits a unique minimal nontrivial normal subgroup H.

b) Prove that a representation of G is faithful if its restriction to H is nontrivial. c) Let $\pi$ be a faithful representation of minimal degree; there exist subgroups $H_i$ of G and representations $\sigma_i$ of $H_i$ of degree 1 such that $\pi$ is isomorphic to $\bigoplus$ Ind$^G_{H_i}(\sigma_i)$. Let $\pi_0=\bigoplus$ Ind$^G_{H_i}(\varepsilon_{H_i})$, where $\varepsilon_{H_i}$ denotes the unit representation. Prove that the kernel of $\pi_0$ is commutative and nontrivial (observe that $\pi_0$ is irreducible); deduce that G is solvable.