Character theory exercises

Question

I'm doing the exercises from chapter 2 of M.Isaacs' Character theory of finite groups, and I'm having problems with some of them.

In particular, I would need help with these ones. Thank you very much in advance!! :)

(2.8) Let $\chi$ be a faithful character of a group $G$. Show that $H\subseteq G $ is abelian if and only if every irreducible constituent of $\chi_{H} $ is linear.

(2.12) Let $|G|=n$ and let $g\in G$. Show that $\chi(g)$ is rational for every character$\chi$ of $G$ if and only if $g$ is conjugate to $g^m$ for every integer $m$ with $(m,n)=1$

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

Answer 1

Hints.

1. Use that $|G/G'|$ is the number of linear characters (this is Corollary 2.23(b)) and consider the number of conjugacy classes in an abelian group.

2. I answered the second question here.

3. Use Lemma 2.27.