Units of integral group rings of Frobenius groups

Let G be a finite Frobenius group with Frobenius kernel N and a Frobenius c omplement X. We prove that if u is an element of U(1)ZG is a torsion unit t hen the order of u divides either \N\ or \X\. As a consequence we prove tha t Zassenhaus' Conjecture holds in some cases and that Problem 8 of [12] has a positive answer for finite groups that are subgroups of the multiplicati ve group of a division ring and for a large family of Frobenius groups. Mor eover, we prove that normalized group bases in the integral group ring of a Frobenius group are Frobenius groups.