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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 $C_{G}(H)=Z(G)$

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closed as too localized by Alexander Gruber, tomasz, Micah, Martin Sleziak, YACP Mar 22 '13 at 22:09

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In the future please ask questions separately with titles descriptive of their content. It will also help if you show us what you tried so we know what part you don't understand. –  Alexander Gruber Mar 22 '13 at 20:54
    
For more about posting multiple questions in one, see this post on meta. –  Martin Sleziak Mar 22 '13 at 22:07

1 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.

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