Zassenhaus Conjecture on torsion units holds for PSL(2,p) with p a Fermat or Mersenne prime

Abstract

Abstract H.J. Zassenhaus conjectured that any unit of finite order in the integral group ring Z G of a finite group G is conjugate in the rational group algebra Q G to an element of the form ± g with g ∈ G . This is known to be true for some series of solvable groups, but recently metabelian counterexamples have been constructed. The conjecture is still open for non-abelian simple groups and has only been proved for thirteen such groups. We prove the Zassenhaus Conjecture for the groups PSL ( 2 , p ) , where p is a Fermat or Mersenne prime. This increases the list of non-abelian simple groups for which the conjecture is known by probably infinitely many, but at least by 50, groups. Our result is an easy consequence of known results and our main theorem which states that the Zassenhaus Conjecture holds for a unit in Z PSL ( 2 , q ) of order coprime with 2 q , for some prime power q .