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abstract: 'Let $G$ be a finite group with a Sylow $p$-subgroup $P$. We prove that the principal $p$-blocks of $G$ and $N_G(P)$ are perfectly isometric under the assumption $G$ has a cyclic $p$-hyperfocal subgroup.'
author:
- Hiroshi HORIMOTO and Atumi WATANABE
title: 'On a perfect isometry between principal $p$-blocks of finite groups with cyclic $p$-hyperfocal subgroups'
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Introduction and preliminaries
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Let $G$ be a finite group and $p$ be a prime number. We denote by $O^{p}(G)$ the subgroup of $G$ generated by the set $G_{p'}$ of $p$-regular elements of $G$. Let $P$ be a Sylow $p$-subgroup of $G$ and set $$\tilde{P}  = \langle \ [\ T, \ O^{p}(N_{G}(T)) \ ] \ | \ T \leq P \ \rangle.$$ We will call $\tilde{P}$ a $p$-hyperfocal subgroup of $G$. By Puig [@Pu5] and [@Pu6], $$\tilde{P} = P \cap O^{p}(G)$$ (see [@BCGLO], Lemma 2.2 or [@Craven], Theorem 1.33 for proofs). Denoting by $b(G)$ the principal block of $G$, $\tilde{P}$ is a hyperfocal subgroup of $b(G)$ in the sense of Puig ([@Pu5] and [@Pu6]). In this paper we prove the following theorem. This study is motivated by Rouquier [@Rou2], A.2.

If $\tilde{P}$ is cyclic, then the principal blocks $b(G)$, $b(N_{G}(\tilde{P}))$ and $b(N_{G}(P))$ 