Units in blocks of defect 1 and the Zassenhaus conjecture

Abstract 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 $$\mathbb {Z}_p G$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub> <mml:mi>Z</mml:mi> <mml:mi>p</mml:mi> </mml:msub> <mml:mi>G</mml:mi> </mml:mrow> </mml:math> 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 $$\mathbb {Z}G$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>Z</mml:mi> <mml:mi>G</mml:mi> </mml:mrow> </mml:math> of order p is conjugate to an element of $$\pm G$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo>±</mml:mo> <mml:mi>G</mml:mi> </mml:mrow> </mml:math> within $$\mathbb {Q}G$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>Q</mml:mi> <mml:mi>G</mml:mi> </mml:mrow> </mml:math> . This is a special case of the Zassenhaus conjecture. We also prove some new results on units of finite order in $$\mathbb {Z}{{\text {PSL}}}(2,q)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>Z</mml:mi> <mml:mtext>PSL</mml:mtext> <mml:mo>(</mml:mo> <mml:mn>2</mml:mn> <mml:mo>,</mml:mo> <mml:mi>q</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> for certain q , and construct a unit of order 15 in $$V(\mathbb {Z}_{(3,5)}{{\text {PSL}}}(2,16))$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>V</mml:mi> <mml:mo>(</mml:mo> <mml:msub> <mml:mi>Z</mml:mi> <mml:mrow> <mml:mo>(</mml:mo> <mml:mn>3</mml:mn> <mml:mo>,</mml:mo> <mml:mn>5</mml:mn> <mml:mo>)</mml:mo> </mml:mrow> </mml:msub> <mml:mtext>PSL</mml:mtext> <mml:mrow> <mml:mo>(</mml:mo> <mml:mn>2</mml:mn> <mml:mo>,</mml:mo> <mml:mn>16</mml:mn> <mml:mo>)</mml:mo> </mml:mrow> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> 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.