Units in Blocks of Defect 1 and the Zassenhaus Conjecture

Building on previous work by Caicedo and the second author, we develop a method that decides the existence of units of finite order in blocks of ${\mathbb{Z}}_{p}G$ of defect 1. This allows us to prove that if $p$ is a prime and $G$ is a finite group whose Sylow $p$-subgroup has order $p$, then any unit $u\in \mathbb{Z}G$ of order $p$ is conjugate to an element of $\pm G$. This is a special case of the Zassenhaus conjecture. We also prove some new results on units of finite order in $\mathbb{Z}\mathrm{P}\mathrm{S}\mathrm{L}(2,q)$ for certain $q$, and construct a unit of order $15$ in $V({\mathbb{Z}}_{(3,5)}\mathrm{P}\mathrm{S}\mathrm{L}(2,16))$ which is a $3$- and $5$-local counterexample to the Zassenhaus conjecture, raising the hope that our methods may lead to a global counterexample amongst simple groups.