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 $$\mathbb {Z}_p G$$ 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 $$\mathbb {Z}G$$ Z G of order p is conjugate to an element of $$\pm G$$ ± G within $$\mathbb {Q}G$$ Q G . This is a special case of the Zassenhaus conjecture. We also prove some new results on units of finite order in $$\mathbb {Z}{{\text {PSL}}}(2,q)$$ Z PSL ( 2 , q ) for certain q, and construct a unit of order 15 in $$V(\mathbb {Z}_{(3,5)}{{\text {PSL}}}(2,16))$$ V ( Z ( 3 , 5 ) 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.