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Let $G$ be a finite group, $p$ a prime, $P \in Syl_p(G)$ with $|P| = p$ and $C_G(P) = P$. Let $H = N_G(P)$. Assume $H/P \cong C_{p-1}$.

Let $\psi \in \text{Irr}(G)$ such that $\psi_H = m \mu$, $m \in \mathbb{N}, \mu \in \text{Irr}(H)$ and $\mu(1) = p-1$. From Brauer's characterization we know $m = 1$.

Is there an "elementary" proof of this result without having to quote Brauer's theory or Dade's theory on cyclic defect groups?

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  • $\begingroup$ This would seemingly follow from a proof of the observation that in your setup, the character values of irreps of $G$ on the conjugacy class of a nonzero element of $P$ are all $0$, $1$ or $-1$. I would be curious to see if this has an elementary proof too. $\endgroup$
    – Chris H
    Commented Jan 9, 2022 at 11:10
  • $\begingroup$ Hi Chris, I agree with your statement. It is easy to deduce that the character values on $p$-elements are rational, but is there an "elementary" way to argue that the values can only be 0, 1 or -1 for irreducible characters as you stated? $\endgroup$
    – Nick
    Commented Jan 9, 2022 at 14:40
  • $\begingroup$ Hi Nick, I don’t know of any proof actually, elementary or not, but my background isn’t very strong. It also seems that rational classes of $G$ having character values equal to the character degrees of the centraliser (up to sign) occur quite frequently, not just in this cyclic case, but I don’t know of any precise statements along these lines. $\endgroup$
    – Chris H
    Commented Jan 9, 2022 at 21:18

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