the second and third main theorem on principal blocks, if $\chi \in {\rm Irr}(b(G))$ then $$\chi * \eta =  \sum_{u \in \Pi }\eta(u)\chi^{(u, b(C_{G}(u)))}$$ is a generalized character belonging to $b(G)$ (c.f. [@BP]).

Let $\tilde{{\cal X}}$ be a set of representatives for the $E$-conjugacy classes of non-trivial linear characters of $\tilde{P}$. For each $\mu \in \tilde{{\cal X}}$, we set $$\eta_{\mu} = 
\sum_{a \in E}\mu^a = \sum_{a \in N_{\tilde{G}}(\tilde{P})/C_{\tilde{G}}
(\tilde{P})}\mu^a.$$ This is $\tilde{G}$-stable.

Suppose that a finite group $G$ has a cyclic Sylow $p$-subgroup $P$. Then the trivial character $1_G$ is not an exceptional character.

We use [@Dor], Theorem 68.1 (cf. [@D]). Set $e = |N_{G}(P): C_{G}(P)|$ and $m  = \frac{|P| -1 }{e}$. Then $e | (p - 1)$. If $m = 1$, then any irreducible character in $b(G)$ is not exceptional. So we may assume $m \neq 1$. Assume that $1_{{G}}$ is exceptional. For some non-trivial linear character $\mu$ of $P$, $$1 = 1_G(\pi) = \varepsilon \sum_{x \in N_{G}(P)/C_{G}(P)} {\mu}(\pi^x) \ (\forall \pi \in {P} \backslash \{ 1 \}),$$ where $\varepsilon = \pm 1$. Then we have $\varepsilon m = \sum_{\pi \in {P} \backslash \{ 1 \}} \mu(\pi) = -1$. This is a contradiction. $\blacksquare$

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[(Dade [@D])]{} ${\rm Irr}(b(\tilde{G})) 