Zassenhaus conjecture on torsion units holds for SL(2, ^𝑝) and SL(2, 𝑝²)

Abstract

H. J. Zassenhaus conjectured that any unit of finite order and augmentation 1 in the integral group ring {\mathbb{Z}G} of a finite group G is conjugate in the rational group algebra {\mathbb{Q}G} to an element of G. We prove the Zassenhaus conjecture for the groups {\mathrm{SL}(2,p)} and {\mathrm{SL}(2,p^{2})} with p a prime number. This is the first infinite family of non-solvable groups for which the Zassenhaus conjecture has been proved. We also prove that if {G=\mathrm{SL}(2,p^{f})} , with f arbitrary and u is a torsion unit of {\mathbb{Z}G} with augmentation 1 and order coprime with p, then u is conjugate in {\mathbb{Q}G} to an element of G. By known results, this reduces the proof of the Zassenhaus conjecture for these groups to proving that every unit of {\mathbb{Z}G} of order a multiple of p and augmentation 1 has order actually equal to p.