we call it the inertial index of $u$.

Irreducible characters in $b(G)$
================================

Under our assumption, let $u \in P$. Then $P_u = C_{P}(u)$ is a Sylow $p$-subgroup of $C_{G}(u)$ and $e_u = |C_{N}(u) : P_u C_{N}(P_u)|$.

There exists $a \in G$ such that $P \cap C_{G}(u^a)$ is a Sylow $p$-subgroup of $C_{G}(u^a)$. Then $u^a \in P \cap C_{G}(u^a) \subseteq P$. By Proposition 2, there exists $n \in N$ such that $u^a = u^n$. Since $P \cap C_{G}(u)^n$ is a Sylow $p$-subgroup of $C_{G}(u)^n $, $P_u = P \cap C_{G}(u) = P^{n^{-1}} \cap C_{G}(u) $ is a Sylow $p$-subgroup of $C_{G}(u)$.

By Proposition 2, we have $N_{G}(P_u) = C_{G}(P_u)(N_{G}(P_u) \cap N)$, hence $$\begin{aligned}
C_{G}(u) \cap N_{G}(P_u) &=& C_{G}(P_u)(C_{G}(u) \cap N_{G}(P_u) \cap N)\\
 &=&  C_{G}(P_u)(C_{G}(u) \cap N). \end{aligned}$$ On the other hand we have $C_{G}(u) \cap C_{G}(P_u)P_u = C_{G}(P_u)P_u$. Therefore $$\begin{aligned}
 e_u &=& |C_{G}(u) \cap N_{G}(P_u) : (C_{G}(u) \cap C_{G}(P_u))P_u|\\
&=& |C_{G}(P_u)(C_{G}(u) \cap N) :  C_{G}(P_u)P_u|\\
&=&|C_{G}(u) \cap N : C_{G}(u) \cap N \cap C_{G}(P_u)P_u| \\
&=& |C_{N}(u) :  N \cap C_{G}(P_u)P_u| \\
&=& |C_{N}(u) : C_{N}(P_u)P_u|. \end{aligned}$$$\blacksquare$

The following is special case of [@Wata2014] Lemma 4 (i). For the self-containedness we state it with a proof because this is crucial.

