On a perfect isometry between principal $p$-blocks of finite groups with cyclic $p$-hyperfocal subgroups

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.