Fusion systems on metacyclic 2-groups

Abstract

Let $P$ be a finite metacyclic $2$-group and $\mathcal{F}$ a fusion system on $P$. We prove that $\mathcal{F}$ is nilpotent unless $P$ has maximal class or $P$ is homocyclic, i.e. $P$ is a direct product of two isomorphic cyclic groups. As a consequence we obtain the numerical invariants for $2$-blocks with metacyclic defect groups. This paper is a part of the author's PhD thesis.