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 \(\mathbb {Z}_p G\) ${\mathbb{Z}}_{p}G$ 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 of \(\mathbb {Z}G\) $\mathbb{Z}G$ of order p is conjugate to an element of \(\pm G\) $\pm G$ within \(\mathbb {Q}G\) $\mathbb{Q}G$ . This is a special case of the Zassenhaus conjecture. We also prove some new results on units of finite order in \(\mathbb {Z}{{\text {PSL}}}(2,q)\) $\mathbb{Z}\text{PSL}(2,q)$ for certain q, and construct a unit of order 15 in \(V(\mathbb {Z}_{(3,5)}{{\text {PSL}}}(2,16))\) $V({\mathbb{Z}}_{(3,5)}\text{PSL}(2,16))$ which is a 3- and 5-local counterexample to the Zassenhaus conjecture, raising the hope that our methods may lead to a global counterexample among simple groups.