Exercise 2.15 M.Isaacs' Character theory of finite groups

Question

I'm beggining to study character theory, and i'm doing some problems from Isaacs' Character theory book.

I would need some help with this one:

(2.15): Let $\chi\in \operatorname{Irr}(G)$ be faithful and suppose $H\subseteq G$ and $\chi_{H} \in \operatorname{Irr}(H)$. Show that $C_{G}(H)=Z(G)$

I have only this... I know i have to use lemma 2.27.
As $\chi$ is faithful, $\ker(\chi)=\{1\}$. From lemma 2.27, we could get that $Z(\chi)=Z(\chi)/\ker(\chi)=Z(G/\ker(\chi))=Z(G)$ and that $Z(G)$ is cyclic. But i'm stuck here.

Thank you very much for any help.

Answer 1

Hint: Schur's Lemma. What do you know about the which commute with (the images of) all elements of $H$ in the associated (irreducible) representation of H?