Exercise 10, § 21

Let $(M, \pi )$ be an irreducible representation of G and H a normal subgroup of G. a) Suppose that H is equal to the center of G. Prove that the degree of $\pi$ divides the index of H in G. (For $m\geqslant 1$, let $H_m$ be the subgroup of $H^m$ consisting of the elements $(h_1, . . . , h_m)$ such that $h_1. . . h_m= 1$. Observe that the representation $\pi^{\times m}: G^m\rightarrow$ End$_K(M^{\otimes m})$ defines an irreducible representation of $G^m/H_m$, and apply Corollary 2 of Proposition 10 (VIII, p. 420) to it.)

b) Suppose that H is commutative. Prove that the degree of $\pi$ divides (G : H). (Using Exercise 9, a), and reasoning by induction on the cardinal of G, reduce to the case when the restriction of $\pi$ to H is isotypical. Prove that $\pi (H)$ is central in $\pi$(G), and apply a).)