Exercise 11, § 21

Let A and H be subgroups of G. Suppose that A is commutative and normal and that G is the semidirect product of H and A. The group H acts on $\widehat{A} =$ Hom(A$,K^*$); choose a system of representatives $(\alpha_i)_{i\in\widehat{A}/H}$ of the orbits of H in $\widehat{A}$. For $i\in \widehat{A}/H$, let $H_i$ be the stabilizer of $\alpha_i$ in H, and let $G_i=$ AH$_i$. For every irreducible representation $\sigma$ of $H_i$, define a representation $\rho_{i,\sigma}$ of $G_i$ by setting $\rho_{i,\sigma}(ah) =$ $\alpha_i(a)\sigma (h)$ for $a\in A$ and $h\in H$. Prove that the representations Ind$^G_{G_i}(\rho_{i,\sigma})$, for $i\in \widehat{A}/H$ and $\sigma \in \widehat{H}_i$, are irreducible and pairwise nonisomorphic, and that we obtain all irreducible representations of G this way (apply Exercise 9).